Lectures in Knot Theory: An Exploration of Contemporary Topics (Universitext) [2024 ed.] 3031400437, 9783031400438

Citation preview

Universitext

Józef H. Przytycki · Rhea Palak Bakshi · Dionne Ibarra · Gabriel Montoya-Vega · Deborah Weeks 

Lectures in Knot Theory An Exploration of Contemporary Topics

Universitext Series Editors Nathanaël Berestycki, Universität Wien, Vienna, Austria Carles Casacuberta, Universitat de Barcelona, Barcelona, Spain John Greenlees, University of Warwick, Coventry, UK Angus MacIntyre, Queen Mary University of London, London, UK Claude Sabbah, École Polytechnique, CNRS, Université Paris-Saclay, Palaiseau, France Endre Süli, University of Oxford, Oxford, UK

Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal, or even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, into very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may find their way into Universitext.

Józef H. Przytycki • Rhea Palak Bakshi • Dionne Ibarra • Gabriel Montoya-Vega • Deborah Weeks

Lectures in Knot Theory An Exploration of Contemporary Topics

123

Józef H. Przytycki Department of Mathematics George Washington University Washington D.C., USA

Rhea Palak Bakshi Institute for Theoretical Studies ETH Zurich Zurich, Switzerland

University of Gdansk Gdansk, Poland Dionne Ibarra School of Mathematics Monash University Melbourne, Australia

Gabriel Montoya-Vega Department of Mathematics The Graduate Center, CUNY New York, NY, USA

Deborah Weeks Department of Mathematics George Washington University Washington D.C., USA

ISSN 0172-5939 ISSN 2191-6675 (electronic) Universitext ISBN 978-3-031-40043-8 ISBN 978-3-031-40044-5 (eBook) https://doi.org/10.1007/978-3-031-40044-5 Mathematics Subject Classification: 57M27, 57M25, 05A30, 16T25, 05C99 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

To my wife, Teresa. To my parents, Bindiya and Jai. To my grandparents, Frances, John, Eva, and Luis. To my grandparents, Jorge Alonso and Amparo del Socorro. To my husband, Andy.

Preface

The initial plan for this book was to publish lecture notes covering several courses in knot theory and low-dimensional topology which the first author taught at the George Washington University over several years, in particular in the academic year 2019–2020. The COVID-19 pandemic partially thwarted these plans, and thus, many lectures in the book are based on notes and surveys of the first author. In particular, Lectures 14, 15, 17, 18, and partially 10 and 20 are based on Mathathons, which are mathematical marathons that have been conducted at GWU every winter for the last 7 years: the first author works with his graduate students for two intensive weeks on some well-formulated open problems, with partial solutions in mind. Mathathons usually take place in December after classes end so as to allow full concentration. All authors of this book participated during the fifth Mathathon, which was held in December 2019 and is discussed in Lecture 15. The topics covered in the book reflect the personal interests of the first author, viewed through the eyes of his Ph.D. students, often with their substantial impact. The use of the book in the classroom very much depends on the personal choice of the instructor and on the course level. However, we suggest that the history of knot theory be covered in the first few classes or perhaps be intertwined with the material of every lecture. For a one semester course we would recommend the theory of skein modules, which is our favorite, and in fact a creation of the first author. This is the first book devoted to this topic to such a great extent. For an undergraduate course we would suggest Lectures 1–10. One can also build a one semester course around Lecture 3. The only background required for most of the book is a basic knowledge of topology and abstract algebra. An introduction to the topology of 3dimensional manifolds is elucidated in Appendices A and B. The book has plenty of exercises, open problems, and conjectures. We hope that the book enables the reader to begin independent research and apply the methods described in the book to other disciplines of science. The writing of this book began in Fall 2019 when all the authors were based at the George Washington University. The first author was partially supported by Simons Collaboration Grant for Mathematicians under grant 3637794. The second author vii

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Preface

acknowledges the support of Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation. The third author was supported by the Swedish Research Council under grant 2021-06594 while in residence at Institut Mittag-Leffler in Djursholm, Sweden, during the winter semester of 2020. The third author further acknowledges support by the Australian Research Council grant DP210103136. The fourth author acknowledges the support of the National Science Foundation through Grant DMS-2212736. Washington, DC, USA Zürich, Switzerland Melbourne, VIC, Australia New York, NY, USA Washington, DC, USA June, 2023

Józef H. Przytycki Rhea Palak Bakshi Dionne Ibarra Gabriel Montoya-Vega Deborah Weeks

Contents

1

2

3

History of Knot Theory from Ancient Times to Gauss and His Student Listing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

From Heraklas to Dürer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Dawn of Knot Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Gauss and the Linking Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.5

Kirchhoff’s Complexity of a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . 10

History of Knot Theory from Gauss to Jones . . . . . . . . . . . . . . . . . . . . . . 15 2.1

Nineteenth-Century Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2

Knot Tabulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3

Tait Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4

Algebraic Topology in Knot Theory . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5

Precision Comes to Knot Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6

The Alexander Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7

Jones Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1

Part I: Fox Coloring, Wirtinger’s Presentation, and Dehn’s Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.1

Introduction to Link Invariants . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.2

Fox 3-Colorings and Generalization to n-Colorings . . . . . . 38 ix

x

Contents

3.2

3.3 4

5

3.1.3

Wirtinger’s and Dehn’s Presentations of the Knot Group . . 43

3.1.4

Constructing Invariants from Magmas: Quandles . . . . . . . . 46

Part II: The Yang-Baxter Operator and Its Homology . . . . . . . . . . . 49 3.2.1

History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2.2

The Set-Theoretic Yang-Baxter Equation . . . . . . . . . . . . . . . 49

3.2.3

From Yang-Baxter Operators to Knot Theory . . . . . . . . . . . 51

3.2.4

Homology of the Yang-Baxter Operator . . . . . . . . . . . . . . . . 54

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Goeritz and Seifert Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2

History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3

Goeritz Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4

Proof of Theorem 4.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5

Signature of Alternating Links and Quasi-alternating Links . . . . . . 72

4.6

Seifert Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.7

The Seifert Form and the Seifert Matrix . . . . . . . . . . . . . . . . . . . . . . 80 4.7.1

Examples of Computing Seifert Matrices from Seifert Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.7.2

S-equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.8

Tristram-Levine Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.9

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

The Jones Polynomial and Kauffman Bracket Polynomial . . . . . . . . . . . 93 5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2

The Jones Polynomial via Skein Relations . . . . . . . . . . . . . . . . . . . . 94

5.3

The Kauffman Bracket Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.4

The Jones Polynomial from the Kauffman Bracket and Their Relation to Fox 3-Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5

The First Tait Conjecture from the Kauffman Bracket Polynomial . 102 5.5.1

Adequate Links and Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 104

5.5.2

Alternating Links Are Adequate . . . . . . . . . . . . . . . . . . . . . . 106

5.5.3

Wu’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.5.4

Turaev’s Construction of TF(D) . . . . . . . . . . . . . . . . . . . . . . . 107

5.5.5

Connected Sum of Alternating Links Is Alternating . . . . . . 108

Contents

xi

5.5.6

6

5.6

Mutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.7

Montesinos-Nakanishi 3-Move Conjecture . . . . . . . . . . . . . . . . . . . . 111

5.8

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

The HOMFLYPT and the Two-Variable Kauffman Polynomial . . . . . 115 6.1

The Alexander Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2

The HOMFLYPT Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3

Criteria for Periodic Links from the HOMFLYPT Polynomial . . . . 122

6.4

Conway Algebras and Entropy Condition . . . . . . . . . . . . . . . . . . . . . 127

6.5

The Two-Variable Kauffman Polynomial . . . . . . . . . . . . . . . . . . . . . . 130 6.5.1

6.6

6.7 7

8

9

The Second Tait Conjecture from the Jones Polynomial and Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

The Dubrovnik Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.6.1

Chebyshev Polynomials as Orthogonal Polynomials . . . . . . 139

6.6.2

Dickson Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Variations on Catalan Connections and the Children Pairing Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.2

Crossingless Connections and Catalan Numbers . . . . . . . . . . . . . . . 143

7.3

Lattice Path and Dyck Path Interpretations of Catalan Numbers . . 152

7.4

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

The Temperley-Lieb Algebra and the Artin Braid Group . . . . . . . . . . . 155 8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8.2

The Temperley-Lieb Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8.3

Temperley-Lieb Algebra’s Connections to Tangles . . . . . . . . . . . . . 158

8.4

The Artin Braid Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8.5

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Symmetrizers of Finite Groups and the Jones-Wenzl Idempotent . . . . 169 9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

9.2

Symmetrizers of Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

xii

10

Contents

9.3

The Jones-Wenzl Idempotent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

9.4

Properties of Tangles Using the Jones-Wenzl Idempotent . . . . . . . . 178

9.5

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Plucking Polynomial of Rooted Trees and Its Use in Knot Theory . . . 185 10.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

10.2

The Plucking Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 10.2.1 q-Commutative Polynomials and Special Plucking Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 10.2.2 Properties of the Plucking Polynomial . . . . . . . . . . . . . . . . . 191

11

10.3

Kauffman Bracket Motivation for the Plucking Polynomial . . . . . . 196

10.4

From Catalan Connections to Rooted Trees . . . . . . . . . . . . . . . . . . . 197

10.5

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Basics of Skein Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 11.1

Introduction to Skein Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

11.2

The Signed Skein Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 11.2.1 q-Deformation of the Fundamental Group . . . . . . . . . . . . . . 208

11.3

The Framing Skein Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

11.4

The Second Skein Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 11.4.1 The q-Homology Skein Module . . . . . . . . . . . . . . . . . . . . . . 212

11.5

The HOMFLYPT Skein Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

11.6

The k-th Skein Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

11.7

The Homotopy Skein Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 11.7.1 q-Analogue of the Homotopy Skein Module . . . . . . . . . . . . 221

11.8

The Kauffman and Dubrovnik Skein Modules . . . . . . . . . . . . . . . . . 223

11.9

The (4, ∞)-Skein Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

11.10 The Bar-Natan Skein Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 11.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 12

The Kauffman Bracket Skein Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 12.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

12.2

The Kauffman Bracket Skein Module . . . . . . . . . . . . . . . . . . . . . . . . 230

12.3

Properties of Kauffman Bracket Skein Modules . . . . . . . . . . . . . . . . 231

12.4

Examples of Kauffman Bracket Skein Modules of 3-Manifolds . . . 236

Contents

13

xiii

12.5

Torsion in Kauffman Bracket Skein Modules . . . . . . . . . . . . . . . . . . 241

12.6

Relative Kauffman Bracket Skein Modules . . . . . . . . . . . . . . . . . . . . 243

12.7

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

The Kauffman Bracket Skein Module and Algebra of Surface I-Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 13.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

13.2

The Kauffman Bracket Skein Module of Surface I-Bundles . . . . . . 250

13.3

The Kauffman Bracket Skein Algebra of a Surface Times an Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

13.4

Properties of Kauffman Bracket Skein Algebras . . . . . . . . . . . . . . . . 256 13.4.1 Connection to the SL(2, C) Character Variety . . . . . . . . . . . 257

13.5

Relative Kauffman Bracket Skein Modules and Algebras of Surface I-Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 13.5.1 Connection to Cluster Algebras . . . . . . . . . . . . . . . . . . . . . . . 261

13.6 14

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened T-Shirt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 14.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 14.1.1 Chebyshev Decoration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

14.2

Structure of the KBSA of the Four-Punctured Sphere . . . . . . . . . . . 268

14.3

The Action of PSL(2, Z) on Multicurves in the Thickened T-Shirt . 270

14.4

Product-to-Sum Formula for Some Families of Curves in the Thickened T-Shirt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 14.4.1 Basic Formulas and Initial Data . . . . . . . . . . . . . . . . . . . . . . . 271 14.4.2 Two Closed Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

15

14.5

The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

14.6

The Positivity Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Spin Structure and the Framing Skein Module of Links in 3-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 15.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

15.2

History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

15.3

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

xiv

16

17

18

19

Contents

15.4

Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

15.5

Spin Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

15.6

Applications to Skein Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

15.7

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

The Witten-Reshetikhin-Turaev Invariants of 3-Manifolds . . . . . . . . . . 295 16.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

16.2

The Witten-Reshetikhin-Turaev Invariant . . . . . . . . . . . . . . . . . . . . . 295

16.3

Calculations of the Witten-Reshetikhin-Turaev Invariant and Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

16.4

The Witten Skein Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

16.5

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

Gram Determinants of Type A and Generalized Type A . . . . . . . . . . . . . 311 17.1

History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

17.2

Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

17.3

The Type A Gram Determinant Formula . . . . . . . . . . . . . . . . . . . . . . 314

17.4

Proof of the Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

17.5

The Generalized Type A Gram Determinant . . . . . . . . . . . . . . . . . . . 318

17.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

17.7

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

Gram Determinants of Type B and Type Mb . . . . . . . . . . . . . . . . . . . . . . . 323 18.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

18.2

The Gram Determinant of Type B . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

18.3

The Gram Determinant Based on the Möbius Band . . . . . . . . . . . . . 327

18.4

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

Khovanov Homology: A Categorification of the Jones Polynomial . . . 335 19.1

History and Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

19.2

The Kauffman Bracket Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 336

19.3

Enhanced Kauffman States and Basis for KH . . . . . . . . . . . . . . . . . . 338

19.4

Translation to Classical Khovanov (Co)homology . . . . . . . . . . . . . 341

Contents

20

xv

19.5

Kauffman Bracket Polynomial as an Euler Characteristic of Khovanov Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

19.6

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

Long Exact Sequence of Khovanov Homology and Torsion . . . . . . . . . . 349 20.1

Long Exact Sequence of Khovanov Homology . . . . . . . . . . . . . . . . 349

20.2

Torsion in Khovanov Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

A Basics of Three-Dimensional Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 A.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

A.2

Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

A.3

Smooth and PL Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 A.3.1 Comparison of the TOP, DIFF, and PL Categories . . . . . . . 373

A.4

Handle Decomposition and Heegaard Decomposition . . . . . . . . . . . 377 A.4.1 Alternative Definitions and the Classification of Lens Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

B

A.5

Prime Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

A.6

Haken Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

A.7

Seifert Fibered Manifolds and JSJ Decomposition . . . . . . . . . . . . . . 388

Surgery on Links in the 3-Sphere and Kirby Calculus . . . . . . . . . . . . . . . 391 B.1

The Mapping Class Group of Surfaces . . . . . . . . . . . . . . . . . . . . . . . 391

B.2

Integral and Dehn Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

B.3

Kirby Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

B.4

The Linking Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

B.5

Parallelizable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 B.5.1 Bundles and Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . 413 B.5.2 Stably Parallelizable and Parallelizable Manifolds . . . . . . . 418

C Table of Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 Select Hints and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 Index of Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

Lecture 1 History of Knot Theory from Ancient Times to Gauss and His Student Listing

Humanity’s fascination with knots can be traced back to prehistoric times. We explore the early appearances of knots and present a chronological overview of the developments that contributed to the formalization of the theory. As the reader may surmise, ancient Greek mathematics played an important role in the initial stages. Some well-known historical names such as da Vinci, Pacioli, Dürer, Leibniz, Euler, and Gauss also take part in the narrative.

1.1 Introduction In this lecture, the captivating history of ideas that lead up to the development of modern knot theory is presented. Knots have fascinated people from the dawn of human history. We can only wonder what caused a merchant living around 1700 BC in Anatolia and exchanging goods with Mesopotamians to choose braids and knots as his seal sign; we refer to [Col] and [Prz11, Prz20] for an image of a cylinder stamp seal from 1700 BC in Anatolia1 and Fig. 1.1 for another seal with a braid from the same time period. However, it is possible that knots appeared in stamps, cylinders, and seals even before proper writing was developed around 3500 BC. Excavations at Lerna by the American School of Classical Studies, under the direction of Professor J. L. Caskey, discovered two rich deposits of clay seal impressions of knots and links. The second deposit dated from around 2200 BC contains several impressions of knots and links; see Fig. 1.3 [Hea]. Recall that the second phase of the early Bronze Age in Greece lasted from 2500 to 2200 BC, and it was a time marked by a considerable increase in prosperity. There were palaces at Lerna and Tiryns (and probably elsewhere) in contact with the second city of Troy. The 1 On the orthogonal base of the hammer-handled seal, from the previous picture, there are patterns surrounding a hieroglyphic inscription (largely erased). Four of the sides are blank and the other four are engraved with elaborated patterns typical of the period, very popular in Syria. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_1

1

2

1 History of Knot Theory from Ancient Times to Gauss and His Student Listing

Fig. 1.1: Modern impression of cylinder seal: royal figures approaching weather god; divinities, from the Old Syrian period ca. 1720–1650 BCE. Image from [Met]

Fig. 1.2: Diagram of a knot represented on a seal from Ur in Mesopotamia

end of this phase was brought by invasions and mass burnings, and the invaders are believed to be the first speakers of the Greek language to arrive in Greece [Prz11]. An even older example of a cylinder seal impression (c. 2600–2500 BC) from Ur, Mesopotamia, is described in the book Inanna by Diane Wolkstein and Samuel Noah Kramer [WK]; the figure there is illustrating the text: “Then a serpent who could not be charmed made its nest in the roots of the tree”2 (Fig. 1.2).

1.2 From Heraklas to Dürer It is tempting to look for the origins of knot theory in ancient Greek mathematics (if not earlier), and there is some justification to do so: a Greek physician named Heraklas, who lived during the first century A.D. and who was likely an associate of Heliodorus, wrote an essay on surgeon’s slings in which he explains, giving step-bystep instructions, 18 ways to tie orthopedic slings. Heraklas’ work survived because Oribasius of Pergamum (ca. 325–400; physician of the emperor Julian the Apostate) 2 Cylinder seal. Ur, Mesopotamia. The Royal Cemetery, Early Dynastic period, c. 2600–2500 B.C. Lapis lazuli. Iraq Museum. An image of the seal can be found in [WK, Prz20, Prz25]. The knot diagram on the seal is the diagram of the grid graph .D(G4,8 ); see Lecture 2, Exercise 2.4.2, and Figs. 2.5, 2.6, and 2.7.

1.2 From Heraklas to Dürer

3

Fig. 1.3: A seal impression from the House of Tiles in Lerna. Image courtesy of the Trustees of the American School of Classical Studies at Athens

included it toward the end of the fourth century in his Medical Collections. The oldest extant manuscript of medical collections was made in the tenth century by the Byzantine physician Nicetas, and it was brought to Italy in the fifteenth century by an eminent Greek scholar, J. Lascaris, a refugee from Constantinople. Heraklas’ part of the Codex of Nicetas had no illustrations, and around the year 1500, an anonymous artist depicted Heraklas’ knots in one of the Greek manuscripts of Oribasius Medical Collections. Figure 1.4 reproduces the third Heraklas knot together with the original Heraklas’ description. Then, Vidus Vidius (1500–1569), a Florentine who became the physician of Francis I (King of France, 1515–1547) and professor of medicine at the Collège de France, translated the Codex of Nicetas to Latin. The translation contains drawings of Heraklas’ surgeons’ slings by the Italian painter, sculptor, and architect Francesco Primaticcio (1504–1570); [Day, Rae]. “For the tying of the crossed noose, a cord, folded double, is procured, and the ends of the cord are held in the left hand, and the loop is held in the right hand. Then the loop is twisted so that the slack parts of the cord crossed. Hence the noose is called crossed. After the slack parts of the cord have been crossed, the loop is placed on the crossing, and the lower slack part of the cord is pulled up through the middle of the loop. Thus the knot of the noose is in the middle, with a loop on one side and two ends on the other. This likewise, in function, is a noose of unequal tension” [Day]. Heraklas’ essay is very important as far as knot theory is concerned even if it is not pure theory but rather its application. The story of survival of Heraklas’ works and efforts to reconstruct his knots in the Renaissance period are typical of all science disciplines and efforts to recover lost Greek books. Mathematics, in general, is a great example of such situation: the beginning of modern calculus in the seventeenth century can be traced back to the reconstruction of books by Archimedes and other ancient Greek mathematicians. It was only the work of Newton and Leibniz that went much further than their Greek predecessors. Proceeding with our journey in

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1 History of Knot Theory from Ancient Times to Gauss and His Student Listing

Fig. 1.4: The crossed noose. Quipus and Witches’ Knots: The Rose of the Knot in Primitive and Ancient Cultures by Cyrus Lawrence Day, published by the University Press of Kansas, ©1967. www.kansaspress.ku.edu. Used by permission of publisher

knot theory history, there are two enlightening examples of great Renaissance artists interested in knots: Leonardo da Vinci (1452–1519) [MaCu] and Albrecht Dürer (1471–1528) [Dür2, Har]. According to Giorgio Vasari [Vas], “da Vinci spent much time in making a regular design of a series of knots, so that the cord may be traced from one end to the other, the whole filling a round space.” Another great artist was Dürer who became acquainted by da Vinci. It does not seem likely that they ever met, but Dürer may have been brought into relation with da Vinci by Luca Pacioli, the author of the book De Divina Proportione which appeared at Venice in 1509 and an intimate friend of Leonardo. Figure 1.5 illustrates knots drawn by these artists.

(a) Knotting by da Vinci. Image from [daV]

(b) Sixth knot by Dürer. Image from [Dür1]

Fig. 1.5: Knots in the Renaissance

1.3 Dawn of Knot Theory

5

1.3 Dawn of Knot Theory We could argue that modern knot theory has its roots with Gottfried Wilhelm Leibniz’s (1646–1716) speculation that aside from calculus and analytical geometry, there should exist a “geometry of position” (geometria situs) which deals with relations depending on position alone, without considering magnitudes. There is evidence in a letter sent to Christiaan Huygens (1629–1695) [Lei], where he declared: “I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with position, as algebra deals with magnitude.” It is not clear whether Leibniz had any convincing example of a problem belonging to the geometry of position. According to [Kli], as far back as 1679, Leibniz in his Characteristica Geometrica tried to formulate basic geometric properties of figures, to use special symbols to represent them, and to combine these properties under operations so as to produce new ones. He called this study analysis situs or geometria situs. Today, we interpret Leibniz’s vision as combinatorial topology. The first convincing example of geometria situs was Leonhard Euler’s (1707–1783) solution to the bridges of Königsberg problem (1735); see Fig. 1.6. The problem was proposed by Heinrich Kühn (1690–1769) around 1735. Kühn was a mathematician born in Königsberg, where he studied at the Pedagogicum, and in 1733 settled in Danzig as a mathematics professor at the Academic Gymnasium (he also was a cofounder of Nature Society and the first person to suggest the geometric interpretation of complex numbers [Jan, Kue]). Kühn communicated the puzzle to Euler, suggesting it may be an example of geometria situs. This communication was made through his friend Carl Leonhard Gottlieb Ehler (1685–1753), the correspondent of Leibniz and future mayor of Danzig.

Fig. 1.6: Bridges of Königsberg problem. Image taken from [Eul] 1736

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1 History of Knot Theory from Ancient Times to Gauss and His Student Listing

The first extant letter by Ehler concerning Königsberg bridges is dated March 9, 1736. There he writes: “You would render to me and our friend Kühn a most valuable service, putting us greatly in your debt, most learned Sir, if you would send us the solution, which you know well, to the problem of the seven Königsberg bridges, together with a proof. It would prove to be an outstanding example of Calculi Situs, worthy of your great genius. I have added a sketch of the said bridges... ” In the reply of April 3, 1736, Euler writes: “... Thus you will see, most noble Sir, how this type of solution bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle. Because of this, I do not know why even questions which bear so little relationship to mathematics are solved more quickly by mathematicians than by others. In the meantime, most noble Sir, you have assigned this question to the geometry of position, but I am ignorant as to what this new discipline involves, and as to which types of problem Leibniz and Wolff expected to see expressed in this way...” [HW]. However, when composing his famous paper on the bridges of Königsberg, Euler had already agreed with Kühn’s suggestion. The geometry of position figures even in the title of the paper Solutio problematis ad geometriam situs pertinentis. There he writes: “The branch of geometry that deals with magnitudes has been zealously studied throughout the past, but there is another branch that has been almost unknown up to now; Leibniz spoke of it first, calling it the geometry of position. This branch of geometry deals with relations depending on position; it does not take magnitudes into considerations, nor does it involves calculations with quantities. But as yet, no satisfactory definition has been given of the problems that belong to this geometry of position or of the method to be used in solving them. Hence, when a problem was recently mentioned, which seemed geometrical but was so constructed that it did not require the measurement of distances—specially as its solution involved only position, and no calculation was of any use. I have therefore decided to give here the method which I have found for solving this kind of problem, as an example...” Euler presented his solution (and generalization) to the bridges of Königsberg problem on August 26, 1735, to the Russian Academy at St. Petersburg (it was submitted for publication in 1736) [Eul]. With this paper, graph theory and topology were born. For the birth of knot theory, one had to wait for another 35 years. In 1771, Alexandre-Theophile Vandermonde (1735–1796) wrote the paper Remarques sur les problemes de situation (Remarks on problems of positions) where he specifically places braids and knots as a subject of the geometry of position [Van]. In the first paragraph of the paper, Vandermonde wrote: “Whatever the twists and turns of a

1.4 Gauss and the Linking Number

7

system of threads in space, one can always obtain an expression for the calculation of its dimensions, but this expression will be of little use in practice. The craftsman who fashions a braid, a net, or some knots will be concerned, not with questions of measurement, but with those of position: what he sees there is the manner in which the threads are interlaced.” Figure 1.7 illustrates some knotting drawn by Vandermonde.

Fig. 1.7: Knotting of Vandermonde. Image from [Van] 1771

1.4 Gauss and the Linking Number In our search for the origin of knot theory, we arrive at the work of Carl Friedrich Gauss (1777–1855). According to [Sta, Dun], one of the oldest notes found among his work after his death is a sheet of paper dated 1794. It bears the heading A Collection of Knots and contains 13 neatly sketched views of knots with English names written beside them. With it are two additional pieces of paper with sketches of knots. One is dated 1819; the other is much later. For instance, Fig. 1.8 shows the meshing knot, the tenth knot of Gauss from 1794.

Fig. 1.8: Tenth knot of Gauss from 1794. In July 1995, the first author visited the old library in Göttingen, where he obtained permission to use the pictures of Gauss’ knots There are other fascinating drawings in Gauss’ notebooks, some of them even suggesting methods for coding a knot, done between 1814 and 1830. For instance, Fig. 1.9 illustrates a method of coding a knot by using a braid.

8

1 History of Knot Theory from Ancient Times to Gauss and His Student Listing a

b

c

d

1

Veraindrunger Coordiniz

2 3 4 5

a

1

1

2+i

3+i

2

2

1

1

2+2i 1

2+2i

b c

3

4

4

4

4

3

d

4

3+i

3+i

2+2i

3+2i

4+3i

1

6

Fig. 1.9: From a notebook of Gauss: “It is a good method of coding a knot” √ As seen in the table above, Gauss used3 the notation i for . −1. Let us attempt to explain Gauss’ coding. Assume the diagram is drawn on the xy-plane while i lies in the z-axis. The four points a, b, c, and d move down along the y-axis. Before passing each crossing, Gauss records the position on the xz-plane by using a complex number. In this way, initially the points a, b, c, and d have x-coordinates 1, 2, 3, and 4, respectively, and coordinate 0 along the z-axis (which means i does not appear). For instance, let us examine the third column which has elements 1, 2, 4, .3 + i. At the first crossing, only c and d switch places. This means a and b remain with x-coordinates 1 and 2, respectively, while c moves to the fourth place and d moves to the third place going down along the z-axis which Gauss denotes as .3 + i. We can presume that Gauss used i to symbolize going down along the z-axis. However, there are some inconsistencies in the table. Nevertheless, it was a very modern approach toward a way of encoding a braid. Exercise 1.4.1 Fully explain how Gauss obtained the table in Fig. 1.9. Are there any mistakes? In another entry in his notebooks (dated January 22, 1833), Gauss introduces the linking number of two knots which is the first deep incursion into knot theory. For example, it can be used to establish that two unlinked circles and the Hopf link are substantially different. In modern language, Gauss’ integral computes the degree of the map from a torus parameterizing a two-component link to the unit sphere.4 It is important to remark that Gauss’ analytical method has been revitalized with Witten’s approach to knot theory [Wit]. James Clerk Maxwell (1831–1879), in his fundamental book of 1873, A Treatise on Electricity and Magnetism [Max], writes: It was the discovery by Gauss of this very integral, expressing the work done on a magnetic pole while describing a closed curve in presence of a closed electrical current, and indicating the geometrical connection 3 In 1572, the Italian mathematician and engineer Rafael √ Bombelli developed the rules for operating with imaginary numbers, introducing the notation i for . −1. He made significant contributions to complex number theory, being praised by renown mathematicians such as Leibniz. 4 Besides electrodynamics, Gauss was also strongly motivated by astronomy. Particularly, he was interested in determining the location of asteroids and in knowing whether their orbits related to that of Earth.

1.4 Gauss and the Linking Number

9

between the two closed curves, that led him to lament the small progress made in the Geometry of Position since the time of Leibniz, Euler, and Vandermonde. We have now, however, some progress to report chiefly due to Riemann, Helmholtz, and Listing. Maxwell goes on to describe two closed curves which cannot be separated but for which the value of the Gauss integral is equal to zero; see Fig. 1.10a.5

(a) The link of Maxwell.

(b) Signs at a crossing.

Fig. 1.10: Gauss linking number In 1876, O. Boeddicker observed that, in a certain sense, the linking number is the number of crossing points of the second curve with a surface bounded by the first curve [Boe1, Boe2, Bog]. Then in 1892, Hermann Karl Brunn [Bru] observed that the linking number of a two-component link can be read from a diagram of the link. If the link has components .K1 and .K2 , we consider any diagram of the link and count each point at which .K1 crosses under .K2 as .+1 or .−1 according to Fig. 1.10b. The sum of these, over all crossings of .K1 under .K2 , is exactly the Gauss linking number. The year 1847 was very important for knot theory (graph theory and topology as well). On the one hand, Gustav Robert Kirchhoff (1824–1887) published his fundamental paper on electrical circuits [Kirc] which had deep connections with knot theory but was discovered a 100 years later (e.g., the Kirchhoff complexity of a circuit corresponds to the determinant of the knot or link determined by the circuit); see Sect. 1.5. On the other hand, Johann Benedict Listing (1808–1882), a student of Gauss, published his monograph Vorstudien zur Topologie (Preliminary Studies in Topology), [Lis]. This monograph has a considerable part devoted to knots, and even earlier, on April 1, 1836, Listing wrote a letter from Catania to Herr Muller, his former school teacher, with the heading “Topology” concerning winding paths of knots and paths in a lattice [Bre1, Bre2, Sta]. He particularly stated that the lefthanded trefoil knot and the right-handed trefoil knot are different. Moreover, he later showed that the figure-eight knot and its mirror image are equivalent. Formally, we say that the figure-eight knot, also called the Listing knot, is amphicheiral. Listing’s indebtedness to Gauss is nicely described in the introduction to the monograph. Here we present a translation to English by Maxim V. Sokolov: “Stimulated by the greatest geometer of our days, who had been repeatedly turning 5 This link is now usually called Whitehead link. However, Maxwell drew and discussed the link decades before Whitehead was even born; [Max].

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1 History of Knot Theory from Ancient Times to Gauss and His Student Listing

my attention to the significance of this subject, during long time I did various attempts to analyze different cases related to the subject, given by natural sciences and their applications. And if now, when these reflections do not have a right yet to claim rigorous scientific form and method, I let myself to publish them as preliminary sketch of the new science, then I do this with the intention to turn attention to the significance and potential of it, with help of collected here important information, examples, and materials. I hope you let me use the name Topology for this kind of studies of spatial images, rather than the name suggested by Leibniz geometria situs... Therefore, by Topology we will mean the study of modal relations of spatial images, or of laws of connectedness, mutual disposition and traces of points, lines surfaces, bodies and their parts or their union in space, independently of relations of measures and quantities...”

1.5 Kirchhoff’s Complexity of a Graph In this section, we develop some aspects of Kirchhoff’s work which are related to knot theory. However, as we mentioned before, the relation with knot theory was not observed until the twentieth century. Gustav Robert Kirchhoff (1824–1887), in his fundamental paper on electrical circuits published in 1847, defined the complexity of a circuit. In the language of graph theory, this complexity of a graph, .τ(G), is the number of spanning trees of G (see Definition 1.5.1 below). It was noted by Tutte around 1936 [BSST] that if e is an edge of G that is not a loop, then .τ(G) satisfies the deleting-contracting relation: .τ(G) = τ(G − e) + τ(G/e), where .G − e is the graph obtained from G by deleting the edge e, and .G/e is obtained from G by contracting e, that is identifying endpoints of e in .G − e. The deleting-contracting relation has an important analogue in knot theory, usually called a skein relation (e.g., Kauffman bracket skein relation; see Lecture 5). Some connections were discovered only about a 100 years later (e.g., the Kirchhoff complexity of a circuit corresponds to the determinant of the knot or link yielded by the circuit). See Lecture 4 for the definition of the determinant of a link. Definition 1.5.1 A spanning tree T of a graph G is defined by the following conditions: (1) T is a subgraph of G. (2) T is a tree. (3) T contains all vertices of G. This is .V (T) = V(G) (spanning condition). Example 1.5.2 For a polygon .Pn , we get .τ(Pn ) = n, and for the graph .Pn obtained from .Pn by doubling one edge, we get .τ(Pn ) = 2n − 1. Exercise 1.5.3 Show that a graph G is a tree if and only if .τ(G) = 1.

1.5 Kirchhoff’s Complexity of a Graph

11

Exercise 1.5.4 Let .G1 ∨ G2 denote a graph obtained from graphs .G1 and .G2 by gluing them along one vertex (.G1 ∨ G2 is not unique; to have uniqueness, we have to specify which vertices are identified). Show that .τ(G1 ∨ G2 ) = τ(G1 )τ(G2 ). Exercise 1.5.5 Consider the family of graphs, which are called the generalized .θcurves. They are graphs of two vertices connected by s paths of length .n1 ,.n2 ,...,.ns , respectively; see Fig. 1.11. Find the number of spanning trees of the first three graphs of Fig. 1.11.

.. .

.. . . . .. . .

.. .

Fig. 1.11: Generalized .θ-curve graphs, .θ 1,1,1 ,.θ 3,1,1 , .θ 4,3,2 , and .θ n1,n2,...,ns

Exercise 1.5.6 Find a closed formula for the number of spanning trees of the generalized .θ-curve .θ n1,n2,...,ns . Exercise 1.5.7 Consider a family of graphs .Gi , .i = 1, ..., s, with two different chosen vertices .vi and .vi. Let .G  be a graph obtained from G by identifying .vi with .vi (we write .G  = G/(vi = vi)). Finally, we define G as the graph obtained from    .(G 1  G 2  ...  G s by identifying .v1 = v2 = ... = vs and .v = v = ... = vs . Show 1 2 the following: (1) For .s = 2, we have the following formula: .

τ(G) = τ(G1 )τ(G2 ) + τ(G1 )τ(G2 ).

(2) Find a formula for .τ(G) for general s. Compare with Exercise 1.5.6. Hint: One can use induction on the number of edges of the graph, starting from the case of .s = 2 and .G2 without edges. But for induction, it is useful to know/guess first the final result. One does, however, a proof in one step using the following hint. Show that the number of trees in .Gi , which contains all vertices but exactly one of   .vi or .v , is equal to .τ(G ). Then we see that the formula for .τ(G s ) is: i i .

τ(G s ) =

s 

  τ(G1 ) · · · τ(Gi−1 )τ(Gi )τ(Gi+1 ) · · · τ(G s ).

i=1

Exercise 1.5.8 Let .Wn be the wheel graph of .n + 1 vertices, that is, the cone over .Pn . Find the closed formula for .τ(Wn ); see Fig. 1.12.

12

1 History of Knot Theory from Ancient Times to Gauss and His Student Listing

W

1

W

2

W

W 3

4

W

W 5

6

Fig. 1.12: Wheel graphs

Hint: Chebyshev polynomials (see Lecture 6) will be involved in the formula (see Example 3 of [Prz21]). For completeness, to be able to clearly see connections to the Goeritz matrix in knot theory (Lecture 4), we define a version of the Kirchhoff matrix of a graph, G, the determinant of which is the complexity .τ(G). Definition 1.5.9 Consider a graph G with vertices .{v0, v1, . . . , vn } possibly with multiple edges and loops (however loops are ignored in the definitions that follow). (1) The adjacency matrix of the graph G is the .(n + 1) × (n + 1) matrix . A(G) whose entries, .ai j , are equal to the number of edges connecting .vi with .v j ; we set .vi,i = 0. (2) The degree matrix .Δ(G) is the diagonal .(n + 1) × (n + 1) matrix whose ith entry is the degree of the vertex .vi (loops are ignored). Thus, the ith entry is equal to  n . j=0 ai j . (3) The Laplacian matrix .Q (G) is defined to be .Δ(G) − A(G). Notice that the sum of the rows of .Q (G) is equal to zero and that .Q (G) is a symmetric matrix. (4) The Kirchoff matrix (or reduced Laplacian matrix) .Q(G) of G is obtained from   .Q (G) by deleting the first row and the first column from .Q (G). Theorem 1.5.10 .det(Q(G)) = τ(G). The shortest proof we are aware of is by directly checking that .det(Q(G)) satisfies the deleting-contracting relation, shown below, where e is any edge that is not a loop: .det(Q(G)) = det(Q(G − e)) + det(Q(G/e)).

1.5 Kirchhoff’s Complexity of a Graph

Example 1.5.11 Consider the graph

13

. For this graph, we have:

.

.

As we will soon see (Fig. 2.4 in Lecture 2), the corresponding knot is the figureeight knot. Moreover, we will see in Lecture 4 that an analogous construction for links (via Tait relation of link diagrams and graphs) is given by the Goeritz matrix of a link diagram. Exercise 1.5.12 Find the Kirchhoff matrix of the graph .Pn . Check that its determinant is equal to .2n − 1.

Lecture 2 History of Knot Theory from Gauss to Jones

In 1867, Lord Kelvin, motivated by Tait’s method of producing vortex smoke rings, came up with the vortex atom theory. He hypothesized that atoms were knotted in a substance called ether. At this time, creating a table of the elements was of significant importance to the scientific community, and this theory encouraged Tait to work on the knot classification problem. In this lecture, the origins of knot theory are examined, taking as a starting point the developments that occurred in Scotland associated with the work of Maxwell, Tait, and Kelvin. Special attention is given to the Tait conjectures and the evolution of the classification of knots. Moreover, we stress how progress in other areas, such as algebraic topology, positively influenced the growth of knot theory. In particular, the Alexander polynomial is discussed. Finally, a historical introduction to the Jones polynomial, a landmark in the theory, is presented.

2.1 Nineteenth-Century Foundations Motivated by Gauss’ recently published work, James C. Maxwell exhibited interest on knots and links in his study of electricity and magnetism of 1873 [Max]. Naturally, the origin of the mathematical theory of knots should be closely associated with advances in chemistry and physics. We explore the fascinating roots of the formal theory beginning with the contributions of Helmholtz, Tait, and Thomson, up to the introduction of the Jones polynomial. The nineteenth century represented a very active time in science dedicated to understanding the structure of the universe. In 1858, the German physicist Hermann von Helmholtz (1821–1894) published his famous paper on vortex1 motion Üe1 A vortex, roughly speaking, is an “area” in a fluid where the flow spins around an axis line. This fluid could be air or water, for instance, and the rotation can take a straight or curved shape. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_2

15

16

2 History of Knot Theory from Gauss to Jones

ber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen (On integrals of the hydrodynamic equations that correspond to vortex motions) where he attempted to apply hydrodynamics to electromagnetic phenomena. Indeed, Helmholtz introduced the notions of vortex line and vortex filament. With these concepts, he characterized the interplay of the fluid and vortex taking place at (something that can be interpreted as) an infinitely small closed curve. He was able to prove, in particular, that there was a proportional correspondence between the motion of the vortex lines and the fluid and the stretching of the line elements [Mof]. It is important to remark that the French mathematician and philosopher René Descartes (1596–1650), during the first half of the seventeenth century, advocated the vortex theory aiming to explain the planetary motion and the composition of the space. Roughly speaking, Descartes explained that when the particles of the universe in chaos started to move, the movement was circular as there was no void in nature. He stated that this circular motion (or vortex) created the orbits of the planets around the sun. Moreover, he thought that the universe consisted of three kinds of elements, i.e., air, earth, and fire, which allowed the characterization of any other substance [Des]. Helmholtz’s work became extensively influential when an English translation was published by Peter Guthrie Tait (1831–1901), a physicist at the University of Edinburgh, who specified that his translation was not exact. However, subsequent revisions made by Helmholtz concluded that the translation “may be accepted as representing the spirit of the original.” Motivated by this paper, in early 1867, Tait developed a simple but effective method of producing vortex smoke rings. As quoted by Tait’s assistant and later biographer, Cargill G. Knott2 [Kno], it was when viewing the behavior of the vortex smoke rings in Tait’s laboratory in Edinburgh that Sir William Thomson (1824–1907) (ennobled Lord Kelvin in 1892) was led to the conception of vortex atoms. Kelvin was a professor of natural philosophy at the University of Glasgow whose investigations aimed to explain the atomic structure of all known elements. Essentially, his vortex atom theory stated that atoms were knotted and linked filaments in an ideal substance called ether [Thoms]. Consequently, Thomson’s theory was Tait’s motivation to understand knots. The point of view initially adopted by Tait was that of classifying knots by the number of crossings, convinced that by performing this tabulation, he would be creating a table of the elements. In Tait’s words: “The enormous number of lines in the spectra of certain elementary substances shows that, if Thomson’s suggestion be correct, the form of the corresponding vortex atoms cannot be regarded as very simple” [Tai3]. There is an interesting letter from Maxwell to Tait dated November 13, 1867, which shows that they were sharing ideas. In one of his rhymes, Maxwell wrote (clearly referring to Tait) [Kno, Sil]: 2 Cargill Gilston Knott (1856–1922). He was a Scottish physicist and mathematician who spent part of his career in Japan [Prz19].

2.2 Knot Tabulation

17

Clear your coil of kinkings. Into perfect plaiting, Locking loops and linkings Interpenetrating. Nevertheless, in 1887, an experiment performed by Albert Michelson (1852– 1931), the first American to win the Nobel Prize in science in 1907, and Edward Morley (1838–1923) proved the nonexistence of ether and further motivated a line of research that eventually led to a new atomic model and special relativity. In particular, the discovery of the electron by Joseph J. Thomson (1856–1940) in 1897 and Ernest Rutherford’s (1871–1937) atomic model of 1909 stand out among the most spectacular advances in science of the time. Consequently, knot theory lost traction among physicists. However, mathematicians were already intrigued and fascinated with the mystery of—pun intended—untangling the speculations generated by knots. As we will see throughout the exploration of this account, it is impressive how a theory originating from an unproven hypothesis is full of elegance and has a wide application potential. Experiments conducted during the last 30 years have brought Kelvin’s vortex atom theory to life once again; as string theory develops, some of Kelvin’s ideas, such as vortex rings, Kelvin waves, and vortex crystals, are being reexamined; see, for instance [Ati1, HNP, Cuo].

2.2 Knot Tabulation Initially, Tait was altogether unaware that anything had been written from a scientific point of view about knots [Tai1]. In fact, it was not until he sent a second paper to the British Association, and as a consequence of a hint from Maxwell, that he obtained a copy of Listing’s essay3 Vorstudien zur Topologie (Introductory Studies in Topology). In Tait’s words: “Here, as was to be expected, I found many of my results anticipated, but I also obtained one or two hints which, though of the briefest, have since been very useful to me. Listing does not enter upon the determination of the number of distinct form of knots with a given number of intersections, in fact he gives only a very few forms as examples, and they are curiously enough confined to three, five and seven crossings only; but he makes several very suggestive remarks about the representation of a particular class of “reduced” knots.”

3 In 1883, Tait wrote in obituary in “Nature” after the death of Listing [Tai2]: One of the few remaining links that still continued to connect our time with that in which Gauss had made Göttingen one of the chief intellectual centres of the civilised world has just be broken by the death of Listing... This paper (Vorstudien zur Topologie), which is throughout elementary deserves careful translation into English...

18

2 History of Knot Theory from Gauss to Jones

Reverend Thomas Penyngton Kirkman (1806–1895), in collaboration with Tait, made considerable progress on the enumeration problem of alternating knots (broadly, an alternating knot is a knot that admits a planar projection such that when “traveling the knot” in a specific direction, the crossings alternate between over and under) so that by 1900 there were in existence tables of (prime) knots up to 10 crossings [Tai3, Kirk]. Motivated by Tait’s paper, Charles Newton Little (1858–1923), an American mathematician and civil engineer, then a professor at the University of Nebraska, addressed the task of enumerating nonalternating knots. His results from 1885 to 1899 included tables of nonalternating knots up to 9 and 10 crossings [Lit1, Lit2]. Little’s nineteenth-century tables were believed to be correct for a long time until in 1973, a duplication of a 10-crossing knot was announced. The duplication was found by Kenneth A. Perko, a former undergraduate student of Ralph H. Fox4 at Princeton who later went to Harvard and became a lawyer. Motivated by a question he was asked from a member of his senior thesis jury, about a drawing in the document, he decided to continue studying knot theory and kept communicating with Fox who was trying to find a counterexample of Poincaré’s conjecture, mainly based on the study of noncyclic covering spaces of knots. The communication between Fox and Perko continued, consisting mainly on the calculation of linking numbers. Eventually, Perko realized he had the tools to tackle the classification of the 10-crossing knots by himself. In January 1973, when working on polynomials of some classical knots of 10 crossings, he observed that two knots yielded a fivefold dihedral linking number of . 32 11 , which he remarks was quite unusual. He was able to see that one knot was the mirror image of the other; the diagrams representing the same knot are called the Perko pair; see Fig. 2.1 [Perk1, Perk2]. The notion of mirror image follows the intuition nicely, namely, the mirror image of a link diagram D, denoted by . D, is the same diagram D with the over-passes changed to under-passes. In some cases, links and their mirror images are equivalent and in some cases they are not. However, in the knot tables, the mirror images are not included. The result by Perko took a while to be highly renowned and the duplication found its way into the first edition of Rolfsen’s book [Rol]. It is important to point out that this discovery by Perko serves as counterexample to “ Little’s theorem” that the writhe (signed sum of all crossings of the diagram with the convention as in Fig. 2.20 where . L+ is assigned a plus one, and . L− a minus one) is an invariant. Recall that during the nineteenth century, knot theory was an experimental science, and at the moment, the concept of equivalence of knots was mostly intuitive. Knots were considered to be circles embedded in space up to “natural deformation,” without giving a formal definition. The aforementioned 4 Ralph Hartzler Fox (1913–1973). After receiving his PhD from Princeton University, he worked at the Institute for Advanced Study in Princeton, the University of Illinois, and Syracuse University before returning to Princeton to join the faculty in 1945. Fox’s career is summarized nicely in the book dedicated to his memory [Neu]: “ The influence of a great teacher and a superb mathematician is measured by his published works, the published works of his students, and perhaps foremost, the mathematical environment he fostered and helped to maintain.” Perko wrote his senior thesis under the supervision of Fox.

2.2 Knot Tabulation

19

claim is now known as one of the Tait conjectures and it was assumed to be true by Max Dehn and Poul Heegaard [DH]. Currently, it is customary to say that the invariance of the writhe for reduced alternating diagrams of an alternating knot is a Tait conjecture. For a detailed narrative of this knot theory discovery, the reader is referred to [Perk4, Perk5].

Fig. 2.1: Diagrams of the Perko pair showing the change in the writhe

Little also published a list of alternating knots of 11 crossings. These tables were partially extended in Mary G. Haseman’s 5 doctoral dissertation of 1917. More precisely, she listed all knots of 12 crossings that are equivalent to their mirror images (such knots are said to be amphicheiral knots) [Has1, Has2]. In 1927, James W. Alexander and his student Garland B. Briggs rigorously proved that knots up to 9 crossings in the tables were in fact all distinct, with the exception of only three pairs of knots. The work by these two mathematicians mainly relied on the use of the first homology groups of branched cyclic coverings. The verification of the 9-crossing knots was completed in 1931 (but not published until 1934) by Carl Bankwitz, a student of the German mathematician Kurt Reidemeister (1893–1971). Reidemeister, in his (German) book [Rei2], regarded the unpublished Bankwitz’s work as very important. Following an off-period, the investigations into the tabulation of knots returned with the work of John H. Conway6 (1937–2020). By using a notation he invented, in 1970, Conway enumerated knots up to 11 crossings; he further worked on the enumeration of the nonsplittable links up to 10 crossings, although this classification was not totally accurate as later noted by Perko and Alain Caudron (he got his PhD in 1987 under Laurence Siebenmann (born in 1939) from the University of Paris) [Cau]. In 1976, Perko received from Hale F. Trotter (born 1931), a professor 5 Mary Gertrude Haseman, professor at the University of Illinois, born on March 6, 1889, in Linton, Indiana. Died on April 9, 1979. She was the fifth doctoral student of Charlotte A. Scott at Bryn Mawr College, who was the first British woman (second in the world after the Russian Sofya Kovalevskaya) to be awarded a doctoral degree in mathematics. Moreover, Scott was very important in the development of mathematics education in the United States. 6 The so-called magical genius, Conway’s legacy reaches several branches of science: from group theory, number theory, analysis, and coding theory to physics and philosophy with the famous free will theorem. Conway received his PhD from Cambridge University in 1964 and was a faculty member there until 1984. Then he went to Princeton University where he held the title of John von Neumann Professor until 2013 when he received emeritus status. He was born in Liverpool, UK, and died on April 11 in New Brunswick, NJ, from complications related to COVID-19. As the UCLA mathematician Terence Tao said, Conway is arguably an extreme point in the convex hull of all mathematicians.

20

2 History of Knot Theory from Gauss to Jones

at Princeton, the results of David A. Lombardero’s senior thesis which was based on a draft of Conway’s 1970 tables. After some analysis of the tables and calculation of some invariants, Perko noticed that a couple of knots were missing from Conway’s published tables. Perko and Caudron, a mathematician at the University of Paris working on the enumeration problem, exchanged data on duplications and omissions in Caudron’s and Conway’s 11-crossing knot tables. At the end of the 1970s, Francis Bonahon (born in 1955) and Siebenmann were able to classify most of the tabled knots. However, some knots fell outside the scope of their work. Perko completed the classification by identifying the knots that were not distinguishable by the Alexander polynomial, so that by 1979 there were tables of knots up to 11 crossings. With this first accurate tabulation, the groundwork was laid to take on the enumeration of higher crossing-number knots [Perk4]. The beginning of the 1980s was marked by significant advances in the enumeration of knots and links with the aid of computers. Indeed, Hugh Dowker (1912–1982), a Canadian mathematician, invented a new notation to tabulate knots mainly based on Tait’s method. In cooperation with Dowker, Morwen Thistlethwaite, an English mathematician, designed an algorithm to generate knots using Dowker’s notation implemented on a computer. This was a very successful experiment and complete tables of prime links up to 12 crossings and prime knots up to 13 crossings were obtained by 1982 [Thi1]. These tables remained intact for about the next 10 years until Jim Hoste, one of the most outstanding knot theorists of the last three decades, decided to tackle the problem of knot enumeration while working together with a group of high school students having access to a powerful computer. In 1995, just before the first author published his Polish book Knots: a combinatorial approach to knot theory (see [Prz8]), he asked Hoste about the current status of tabulation. Hoste informed that he and, at first independently, Thistlethwaite were working on the extension of the tables up to 15- and 16-crossing knots. Moreover, an American mathematician named Jeffrey R. Weeks, who was specializing in the area of hyperbolic knots, played a very important role in Hoste’s team. Remarkably, although both teams kept their tabulation progress secret until the tables were complete, they obtained the same results. As Hoste, Thistlethwaite, and Weeks mentioned, their overall programs were similar in spirit, being the main difference the use of hyperbolic geometry by Hoste and Weeks. The paper they published bears the name “The first 1,701,936 knots” [HTW]. Finally, notice that knots and their mirror images are not counted separately. Hoste, Thistlethwaite, and Weeks determined which knots in the table are amphicheiral, so that, for amphicheiral knots, no information is missing, while nonamphicheiral knots in the list actually represent two knots. The next major achievement in the knot classification was made by Benjamin A. Burton, a mathematician at the University of Queensland in Brisbane, Australia. In his paper, the knot tables were extended up to 19 crossings, with a total

2.3 Tait Conjectures

21

of 352,152,252 distinct nontrivial prime knots. Burton’s method implemented a varied use of hyperbolic geometry, knot polynomials, normal surface theory, and computational algebra, and as he mentions, it took the assistance of a few hundred machines. The reader is invited to read [Bur] a further description of the impressive methodology used to generate this tabulation. Burton has also been working on the enumeration up to 21 crossings, but at the moment of writing this book, his work had not been published yet. We finish the section with Table 2.1 summarizing the current information on knot enumeration. As Hoste remarks in [Hos], the numbers of knots with at least 17 crossings should be regarded with “healthy skepticism,” even though independent tabulations have verified the numbers. Number of crossings Alternating Nonalternating 3 1 0 4 1 0 5 2 0 6 3 0 7 7 0 8 18 3 9 41 8 10 123 42 11 367 185 12 1288 888 13 4878 5110 14 19536 27436 15 85263 168030 16 379799 1008906 ∗ ∗ 17 .1769979 .6283414 ∗ ∗ 18 .8400285 .39866181 ∗ ∗ 19 .40619385 .253511073 ∗ 20 .199631989 ? ∗ 21 .990623857 ? ∗ 22 .4976016485 ? ∗ 23 .25182878921 ? Table 2.1: Enumeration of knots

2.3 Tait Conjectures Undoubtedly, Tait’s role was key in the early development of tabulation, specifically of alternating knots. In this section, we present the principles he introduced in order to be able to make the tables of knots. These principles are now called Tait conjectures.

22

2 History of Knot Theory from Gauss to Jones

First, define an alternating diagram of a knot or link to be reduced if it contains no nugatory crossings, as shown in Fig. 2.2. One of the main concerns when classifying knots is to look for the diagram with the minimal number of crossings. The number of crossings in the diagram can be immediately decreased if the diagram has a nugatory crossing. Tait noticed that he was never able to reduce further a reduced alternating diagram. Tait observations are now formulated as follows:

Flype move.

Fig. 2.2: Nugatory crossing: the twist can be removed by rotation of one of the sides of the diagram (left) and a flype move: a rotation of .180◦ along the x-axis (right)

Conjecture 2.3.1 An alternating diagram with no nugatory crossings, of an alternating link, realizes the minimal number of crossings among all diagrams representing the link. Conjecture 2.3.2 Two alternating diagrams, with no nugatory crossings, of the same oriented link have the same writhe number. Conjecture 2.3.3 Two alternating diagrams, with no nugatory crossings, of the same link, are related by a sequence of flypes (now called Tait flypes), as in Fig. 2.2. Several mathematicians attempted to demonstrate the conjectures for over 80 years, but no one was successful. Only until 1986, after the announcement of a new type of knot invariants, the first conjecture was proved. Three mathematicians, i.e., Louis Kauffman, Kunio Murasugi, and Morwen Thistlethwaite, implemented the newly discovered Jones polynomial and Kauffman bracket interpretation to prove the first conjecture [Kau5, Mura1, Thi2]. The second Tait conjecture was proved independently by Murasugi and Thistlethwaite in 1987 [Mura1, Thi3]. Finally, the flype conjecture was proved in 1991 by William Menasco and Thistlethwaite [MeTh]. The reader who desires to further explore the developments in nineteenth-century knot theory is referred to the survey in the Dehn and Heegaard article in the Mathematical Encyclopedia from 1907.7 In particular, they provided the first formal definition of a knot and “isotopy” [DH]. Moreover, in this context, the papers of 7 Max W. Dehn was a German Mathematician (1878–1952) who got his PhD under David Hilbert in 1900. He is famous, among other things, of solving Hilbert’s third problem and Dehn’s surgery. Poul Heegaard (1871–1948) was a Danish mathematician who in his doctoral thesis of 1898 introduced the Heegaard splitting of a 3-manifold.

2.4 Algebraic Topology in Knot Theory

23

Oscar Simony (1852–1915), a mathematician from Vienna, are of great interest specially regarding torus knots and 2-bridge knots [Simo]. Simony was basically using continued fractions (in a similar way as Conway) to describe torus knots. Figure 2.3 shows torus knots as drawn by Simony.

Fig. 2.3: Simony’s knots from 1884

2.4 Algebraic Topology in Knot Theory Historically, the fundamental problem in knot theory is distinguishing nonequivalent knots. Progress was not made until the French mathematician, theoretical physicist, and science philosopher Jules Henri Poincaré (1854–1912), one of the greatest of all times, laid the foundations of algebraic topology in his Analysis Situs paper from 1895 [Poi1]. In particular, he introduced the notion of homotopy and homology groups. What follows is an excursion into the most representative early events that over time established the relation between algebraic topology and knot theory. This eventually led to the construction of the Alexander polynomial. In his doctoral dissertation of 1898 [Hee], Poul Heegaard constructed the twofold branch cover of the trefoil knot, showing that it is the (now known as) lens space . L(3, 1); see Appendix A; he analogously showed that the same process for the unknot yields .S 3 . Heegaard was able to distinguish . L(3, 1) from .S 3 by noting that its first homology group is a nontrivial torsion group. However, he did not state that this fact could be used to distinguish the trefoil knot from the unknot. Then in 1906, the Austrian mathematician Heinrich Friedrich Tietze (1880–1964) used the fundamental group of the exterior of a knot in .R3 , called the knot group, to distinguish the unknot from the trefoil [Tie1]. Recall that the notion of fundamental group was introduced by Poincaré.

According to Dehn’s wife, Mrs. Toni Dehn, Dehn and Heegaard met at the third International Congress of Mathematicians at Heidelberg in 1904 and took to each other immediately. They left Heidelberg on the same train, Dehn going to Hamburg and Heegaard returning to Copenhagen. They decided on the train that an Encyclopedia article on topology would be desirable, that they would propose themselves as authors to the editors, and that Heegaard would take care of the literature, whereas Dehn would outline a systematic approach which would lay the foundations of the discipline [Mag].

24

2 History of Knot Theory from Gauss to Jones

During a lecture delivered at a meeting of the German Mathematical Society in 1905 in Meran, an Austrian mathematician, Wilhelm Wirtinger, (1865–1945), outlined a method of finding a knot group presentation (currently called Wirtinger’s presentation) [Wir]; see Lecture 3. Later Max Dehn constructed a different presentation of the fundamental group of a knot complement independently of Wirtinger (now known as Dehn’s presentation8 ); see Lecture 3. By using his presentation, Dehn in his 1910 paper [Deh2] was able to distinguish the right-handed trefoil knot from its mirror image,9 the left-handed trefoil knot. In other words, Dehn showed that the trefoil knot is nonamphicheiral. As previously mentioned, graph theory and topology have a closely related origin, so it is only natural to study the relation between knots and planar graphs. Tait was the first to notice such a connection. More precisely, he colored the regions of the knot diagram alternately white and black and constructed the graph by placing a vertex inside each white region and then connecting the vertices by edges going through the crossing points of the diagram. Figure 2.4 depicts the process.

Fig. 2.4: From the figure-eight knot diagram to its plane graph

It is useful to mention that Tait also considered the construction in the opposite direction, namely, going from a plane graph G to a link diagram . D(G). In his construction, we replace every edge of a graph by a crossing according to the convention of Fig. 2.5 and connect endpoints along edges as in Figs. 2.6 and 2.7. We should mention here one important observation already known to Tait (and in implicit form to Listing): Exercise 2.4.1 Show that the diagram . D(G) of a plane graph G is always alternating. Hint: See Fig. 2.8. 8 In 1978, W. Magnus wrote [Mag]: “Today, it appears to be a hopeless task to assign priorities for the definition and the use of fundamental groups in the study of knots, particularly since Dehn had announced [Deh1] one of the important results of his 1910 paper (the construction of Poincaré spaces with the help of knots) already in 1907.” 9 By the mirror image of the diagram, we understand the image under reflection on the xy-plane. In this case, every overcrossing is changed to undercrossing and vice versa. Notice that if we take another plane of reflection, we may obtain a different diagram but it is still an equivalent link. A link that is equivalent to its mirror image is said to be amphicheiral.

2.4 Algebraic Topology in Knot Theory

25

Fig. 2.5: Convention for a crossing

10

14 9

Fig. 2.6: From a plane graph to the knot .1014 in [Rol] notation

Fig. 2.7: Octahedral graph and the associated link diagram

Fig. 2.8: Alternating and nonalternating parts of a diagram

Tait realized that not every knot has an alternating diagram [Tai3]. He did not have the tools to prove it but he did sufficient experimental checking to convince himself. His first example was the knot shown in Fig. 2.9 (here a negative edge means that we use the opposite convention to that of Fig. 2.5). Notice that only in the twentieth century the existence of nonalternating knots was proven [Ban, Crow].

26

2 History of Knot Theory from Gauss to Jones

Fig. 2.9: The knot .819 and its Tait graph (.819 in [Rol] notation); it is the first nonalternating knot in tables

Exercise 2.4.2 Draw the Tait diagram of the grid graph .G3,3 = . Notice that the diagram is that of an ancient mosaic as illustrated in Fig. 2.10.

Fig. 2.10: Mosaic flooring from a tetraconch in the courtyard of the Library of Hadrian. The tetraconch is thought to be the first episcopal church in Athens, Greece, and constructed around the second quarter of the fifth century. Image taken by the third author

2.5 Precision Comes to Knot Theory Throughout the nineteenth century, knots were understood to be closed curves in space up to natural deformation. This concept was not formally defined yet; rather, it was described as a “movement in space without cutting and pasting.” Despite not having formal definitions but an experimental intuition, we have seen that scientists such as Tait, Kirkman, Little, and Haseman were able to make important contributions to the theory. However, precise methods allowing to distinguish knots, which could not be practically deformed from one to another, were still absent. In 1907, Max Dehn and Poul Heegaard, in their famous Mathematical Encyclopedia, outlined a systematic approach to topology. In particular, they precisely formulated the subject of knot theory [DH]. To bypass the notion of deformation of a curve in a space, they introduced the concept of lattice knots together with

2.5 Precision Comes to Knot Theory

27

the definition of their equivalence. Later, Reidemeister and Alexander considered more general polygonal knots in a space with equivalent knots related by a sequence of .Δ-moves. Moreover, they also explained .Δ-moves by elementary moves on link diagrams (currently known as Reidemeister moves). The definition of Dehn and Heegaard was long ignored and only recently the concept of lattice knots is being studied again. It is important to mention that the two concepts, lattice knots and polygonal knots, are equivalent.10 In this section, we define, following Reidemeister, a polygonal knot and link and .Δ-equivalence of knots and links. A .Δ-move is an elementary deformation of a polygonal knot which intuitively agrees with the notion of “deforming without cutting and gluing,” which is a historical underlining principle of topology. Definition 2.5.1 (a) A polygonal link is a collection of disjoint simple closed polygonal curves in 3 .R . A polygonal link consisting of one simple closed curve is called a polygonal knot. (b) Let u be a line segment (edge) in a polygonal link L in .R3 . Let .Δ be a triangle in 3 .R whose boundary consists of three line segments denoted by u, v, and w, such that .Δ ∩ L = u. The polygonal curve . L ', defined as . L ' = (L − u) ∪ v ∪ w, is a new polygonal link in .R3 . We say that the link . L ' was obtained from L by a .Δ-move. Conversely, we say that L is obtained from . L ' by a .Δ−1 -move; see Fig. 2.11.

Fig. 2.11: Replacing the edge u, with the two edges v and w, by a .Δ-move

Moreover, the triangle .Δ is allowed to be degenerate so that the vertex .v ∩ w is on the side u. In other words, subdivision of the line segment u is allowed, as depicted in Fig. 2.12.

Fig. 2.12: Any subdivision is a combination of three nondegenerate .Δ-moves

10 It is a long routine exercise, a folklore result.

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2 History of Knot Theory from Gauss to Jones

(c) Two polygonal links are said to be .Δ-equivalent (or combinatorially equivalent) if one can be obtained from the other by a finite sequence of .Δ- and .Δ−1 − moves. Polygonal links are usually presented by their diagrams in a plane. Informally, diagrams are defined using projections that allow us to describe the “over and under” information of a crossing. In order for that information to be well-defined, we must limit our discussion to regular projections of a link. Let . p : R3 → R2 be a projection and let . L ⊂ R3 be a link. Then a point .P ∈ p(L) is called a multiple point of p if −1 −1 . p (P) contains more than one point (the number of points in . p (P) is called the multiplicity of P). Definition 2.5.2 The projection p is called regular if: (1) p has only a finite number of multiple points and all of them are of multiplicity two (2) No vertex of the polygonal link is an inverse image of a multiple point of p Thus, in case of a regular projection, the parts of a diagram illustrated in Fig. 2.13 are not allowed.

Fig. 2.13: Not allowed in a regular projection of a polygonal link

Definition 2.5.3 A link diagram is a regular projection of the link with the “over and under” information at every crossing. Figure 2.14 illustrates a diagram of the trefoil knot. It is important to remark that for practical purposes, when drawing a knot or a link, it is assumed that there is a very large number of edges composing the knot or link, so that the curve seems to be “well rounded.” Maxwell was the first person to consider the question when two projections represent equivalent knots. He considered some elementary moves (later known as Reidemeister moves), but never published his findings. .Δ-moves were used independently by Reidemeister, and Alexander and Briggs,11 to describe equivalence of links using diagrams [Rei1, AB].

11 Alexander and Briggs used the name elementary deformations to refer to the moves.

2.6 The Alexander Polynomial

29

Fig. 2.14: Polygonal trefoil knot

Theorem 2.5.4 (Reidemeister Theorem) Two link diagrams represent .Δequivalent12 links if and only if they are related by a finite sequence of Reidemeister moves, . Ri for .i = 1, 2, 3, and isotopy (deformation) of the plane of the diagram. See Fig. 2.15. The moves are allowed in both directions, as depicted in the figure below. The theorem also holds for oriented link diagrams. In this case, all possible coherent orientations of diagrams involved in the moves must be taken into account.

Fig. 2.15: Reidemeister moves

2.6 The Alexander Polynomial In 1912, the American mathematician George David Birkhoff (1844–1944), when trying to prove the four-color problem (formulated by Francis Guthrie in 1852), introduced the chromatic polynomial of a graph [Bir1]. Loosely speaking, this polynomial counts the number of labels or colorings of vertices of a graph, in such a way that vertices connected by an edge have different colors. A breakthrough in the theory of invariants of knots was the discovery of a Laurent polynomial invariant of knots by James Waddell Alexander (1888–1971). He was an American mathematician born in New Jersey and obtained his PhD in 1915 from Princeton University under the supervision of Oswald Veblen (1880–1960). An excellent climber and 12

In modern knot theory, especially after the work of Fox, we use the equivalent notion of ambient isotopy in .R3 or in .S3 . In that way, two links in a 3-manifold M are said to be ambient isotopic, if there is an isotopy of M sending one link into the other.

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2 History of Knot Theory from Gauss to Jones

mountaineer, he succeeded in some popular ascents such as the Swiss Alps and the Colorado Rockies. Moreover, he became a millionaire through his maternal grandfather, who was the president of the Equitable Life Assurance Society, which is why he did not take a salary while holding teaching positions at Princeton. Alexander’s time coincided with Birkhoff’s time in Princeton, and it can be conjectured that he knew about the chromatic polynomial.13 Alexander gave several approaches to his polynomial. Here we sketch the combinatorial construction of the polynomial using the diagram of the link [Ale4]. Before we describe the construction of this polynomial, it is worth taking a look at an excerpt from a letter that Alexander sent to Oswald Veblen in 1919, where he describes the numerical precursor to his polynomial for the first time [Ale1]. This was only known to a very small group of specialists; we thank Jim Hoste and Peggy Kidwell (widow of Mark Kidwell) for providing a copy of the letter:

When looking over Tait on knots among other things, he really doesn’t get very far. He merely writes down all the plane projections of knots with a limited number of crossings, tries out a few transformations that he happen to think of and assumes without proof that if he is unable to reduce one knot to another with a reasonable number of tries, the two are distinct. His invariant, the generalization of the Gaussian invariant ... for links is an invariant merely of the particular projection of the knot that you are dealing with,—the very thing I kept running up against in trying to get an integral that would apply. The same is true of his “Beknottednes”. Here is a genuine and rather jolly invariant: take a plane projection of the knot and color alternate regions light blue (or if you prefer, baby pink). Walk all the way around the knot and [See Fig. 2.16] every time you go over a crossing, put a dot to either side of the curve just beyond the crossing.

Fig. 2.16: Trefoil knot in a letter from Alexander to Veblen

13 Birkhoff writes in [Bir2]: “...Alexander, then a graduate student, began to be especially interested in the subject (analysis situs). His well known “duality theorem”, his contributions to the theory of knots, and various other results, have made him a particularly important worker in the field.” We can also mention that in the fall of 1909, Birkhoff became a member of the faculty at Princeton and left for Harvard in 1912. His 1912 paper [Bir1] ends with “Princeton University, May 4, 1912.”

2.6 The Alexander Polynomial

31

Form a square matrix with one column to each vertex and one row to each region, except that you leave out one arbitrary blue region and one arbitrary white region (most conveniently the outside one). Put a 0 in the i-th row and j-th column if the jth vertex is not on the border of ith region, a .+1 if the jth vertex is on the boundary of the i-th region, and if there is a dot in the corner of the region corresponding to this vertex. Put a .−1 if the vertex is on the boundary but there is no dot in the corner. The elementary divisors of the matrix are invariants of the knot (or link: the thing goes equally well for links). The number of even elementary divisors is one less than the number of distinct curves of the system. If we build one system up by piercing on other systems . A, B, C Figure [meaning probably connected sum] the determinant of the resultant system is the product of the determinants of the individual ones. More about this and other things when we meet. I am waiting for your book with impatience. Sincerely Alexander P.S. The trefoil knot has the invariant .(3), the Bear design: [See Fig. 2.17] the invariant .(4, 4). There are at least several errors in Tait’s classification.

Fig. 2.17: Final part of the letter

Less than 10 years after sending the letter to Veblen, Alexander came back to this idea and generalized it to a module and a polynomial, now called Alexander module and Alexander polynomial, respectively. Alexander represented a knot schematically as a 2-dimensional figure or diagram calling the singularities of the curve crossing points. He first observed that for a diagram D with n crossings, the arcs divide the projection plane into n+2 regions, including the region “outside” the knot. After an appropriate labeling, each crossing yields an equation in terms of the regions. These equations can be arranged in a .(n × (n + 2))-matrix, whose determinant (after removing two columns) is a polynomial and the Alexander polynomial is obtained after a normalization of this determinant. The process is described as follows. Definition 2.6.1 Let D be a knot diagram.

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2 History of Knot Theory from Gauss to Jones

• Let v be a crossing of D. Assign two dots on the left-hand side of the oriented under strand and label the four regions surrounding the crossing as .r1 , .r2 , .r3 , and .r4 , as in Fig. 2.18. • At each crossing v, take an alternating sum of the .ri ’s staring at .r1 , going around the crossing counterclockwise. Multiply the dotted regions by t. A linear equation in four variables is obtained: .tr1 − tr2 + r3 − r4 = 0. Defining an equation for each crossing results in n equations with .n + 2 variables, which can be represented as an .(n × (n + 2))-matrix, where each entry is .±t, .±1, or 0. Alexander then deleted two columns from the matrix. These columns corresponded to any two neighboring regions, and he later showed that the choice of the columns does not affect the polynomial. The resultant square matrix is the so-called Alexander matrix of the knot represented by the diagram, and its determinant is a polynomial in the variable t. The polynomial depends heavily on several factors: ordering of crossings and labeling of the regions, orientation, choice of columns to be deleted, and most importantly the choice of the knot diagram. Alexander proved that his polynomial is a link invariant up to a factor of .±t k for some integer k. He normalized this polynomial so that it has a positive constant term; the resulting Alexander polynomial is a knot invariant [Ale1]. Let us illustrate the calculation of the Alexander polynomial of the trefoil knot as follows. Consider the diagram as in Fig. 2.18.

Fig. 2.18: Alexander polynomial The following matrix is obtained from the three equations in the variables .r0 , .r1 , r2 , .r3 , and .r4 : t −1 0 −t 1 ( ) . M = }t −t −1 0 1| . (t 0 −t −1 1)

.

By deleting the last two columns (corresponding to the neighboring regions .r3 and .r4 ) and calculating the determinant of the resultant .3 × 3-matrix, the polynomial 2 .(t − t + 1)t is obtained. Thus, the Alexander polynomial denoted by .Δ, of the trefoil knot is given by .Δ = t 2 − t + 1.

2.6 The Alexander Polynomial

33

Alexander uses the following observation in the proof of the existence of his polynomial: Exercise 2.6.2 Consider an oriented link diagram D (or just its projection, as overunder information is not needed). Show that one can always color regions of .R2 − D by integers in such a way that neighboring regions have numbers which differ by one and the unbounded region has color 0. Furthermore, this coloring is unique. In the coloring, we use the convention that if we travel along the diagram according to its orientation, then the bigger number is on the left side (as in Fig. 2.19 (left)). Hint: Notice that by choosing a color for a corner of the crossing and then going around the crossing, we end with that color; see Fig. 2.19 (right) (this is the case because any crossing has two inputs and two outputs).

n n

n+1

n−1

n+1 n

Fig. 2.19: Alexander numbering of regions of .R2 − D Observe that the Alexander numbering modulo 2 yields the checkerboard coloring. It is important to mention that the Alexander polynomial can also be derived from the knot group, that is, the fundamental group of the knot complement in .S 3 , and this point of view has been extensively developed. More generally, under the leadership of Ralph Fox, the study of the fundamental group of a knot complement and branch coverings was the main topic of research in knot theory for the next 50 years. This research trend culminated in 1988 with the proof of Tietze’s conjecture by Gordon and Luecke [GoLu, Tie1]. The conjecture states that a knot is completely determined by its complement [Bir1, Rol]. Furthermore, Alexander observed that if three oriented links . L+ , . L− , and . L0 have diagrams which are identical except near one crossing, as in Fig. 2.20, then their polynomials are linearly related [Ale1].

Fig. 2.20: Three diagrams differing only in the parts shown An analogous discovery for chromatic polynomials was made by Ronald M. Foster in 1932 [Whit]. We finish this section mentioning that in the 1960s, Conway

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2 History of Knot Theory from Gauss to Jones

rediscovered Alexander’s formula and normalized the polynomial .Δ L (t) ∈ Z[t ± 2 ], defining it recursively as [Con1]: 1

1. .Δ O (t) = 1, where .O denotes the knot diagram, and √ 1 2. .Δ L+ − Δ L− = ( t − √ )Δ L0 . t

2.7 Jones Revolution A new polynomial announced in the spring of 1984 was a milestone in knot theory. The New Zealander mathematician Vaughan Frederick Randal Jones (1952–2020) discovered what is now known as the Jones polynomial while working on von Neumann algebras [Jon2, Jon3, Jon4]. The following is an excerpt from [Jon6]: “It was a warm spring morning in May, 1984, when I took the uptown subway to Columbia University to meet with Joan S. Birman,14 a specialist in the theory of braids... In my work on von Neumann algebras, I had been astonished to discover expressions that bore a strong resemblance to the algebraic expression of certain topological relations among braids. I was hoping that the techniques I had been using would prove valuable in knot theory. Maybe I could even deduce some new facts about the Alexander polynomial. I went home somewhat depressed after a long day of discussions with Birman. It did not seem that my ideas were at all relevant to the Alexander polynomial or to anything else in knot theory. But one night the following week I found myself sitting up in bed and running off to do a few calculations. Success came with a much simpler approach than the one that I had been trying. I realized I had generated a polynomial invariant of knots.” Jones denoted the polynomial .VL (t), and soon he realized that his polynomial satisfied the local relations analogous to that discovered by Alexander and Conway: √ 1 t −1VL+ − tVL− = ( t − √ )VL0 , t

.

and established the meaning of .t = −1. He observed that .VL (−1) is equal to the determinant of the knot, a classical knot invariant. The Jones polynomial distinguishes some knots with the same Alexander polynomial; one of the most puzzling questions 14 Joan Sylvia Lyttle Birman, a New Yorker mathematician born May 30, 1927. She received her PhD in 1968 at the Courant Institute at New York University under the direction of Wilhelm Magnus (a student of Dehn) and is recognized as one of the leading researches in low-dimensional topology specially knot theory, 3-manifolds, and mapping class groups of surfaces. Among Birman’s honors we find the very prestigious Chauvenet Prize of the Mathematical Association of America (1996) and an honorary doctorate by the Technion Israel Institute of Technology in 1997. Moreover, she was a member of the Institute for Advanced Study at Princeton in 1987 and she has had 21 doctoral students, including Józef Henryk Przytycki.

2.7 Jones Revolution

35

in the theory is whether the Jones polynomial detects the unknot (now called the Jones conjecture). We refer to Lecture 5 for further discussions on this topic. As expected, this new polynomial triggered a new trend in knot theory research. Amazingly enough, in the summer and fall of 1984, both Alexander and Jones polynomials were independently generalized to a two-variable polynomial by four different groups of mathematicians. This new Laurent polynomial, denoted by .PL ∈ Z[a±1, z ±1 ], carries the name HOMFLYPT, which is the acronym after the initials of the inventors: Jim Hoste, Adrian Ocneanu, Kenneth Millett, Peter Freyd, W. B. Raymond Lickorish, David Yetter, Józef H. Przytycki, and Pawel Traczyk [FHLMOY, PT1].15 The HOMFLYPT polynomial is discussed in detail in Lecture 6. This polynomial is defined recursively, in a similar way as the Alexander and Jones polynomials, as follows: 1. .P O = 1, where .O denotes the trivial knot diagram, and 2. .aPL+ + a−1 PL− = zPLO . In particular, the Alexander polynomial and the Jones polynomial can be obtained from the HOMFLYPT by specific values of a and z, namely: √ 1 Δ L (t) = PL (i, i( t − √ )) and, t

.

√ 1 VL (t) = PL (it −1, i( t − √ )). t

.

In August 1985, Louis H. Kauffman (born in February 1945), an American mathematician, announced a new invariant of unoriented framed links (now called the Kauffman polynomial of two variables); see Lecture 6. As he remarks, almost immediately after his discovery, he observed the bracket state summation as a special case of the Kauffman original two-variable polynomial. Initially, he thought the bracket was an entirely new invariant. However, a day later, he found that the bracket gave a new and simple model for the Jones polynomial [Kau6]. It should be noted, as first observed by Kauffman, that the bracket polynomial in three variables .(A, B, d) (see Sect. 5.3) is an isotopy invariant of alternating links. This was observed under the assumption that the third Tait conjecture holds [MeTh]. The reader will have plenty of opportunities later in this book to explore the far-reaching implications and developments of knot theory closely related to the bracket polynomial. Lecture 5 discusses the Kauffman bracket polynomial and its connection to the Jones polynomial. Invariants of Jones type lead to the rapid development of the theory and generalizations to invariants of 3-manifolds, for example, skein modules and Vassiliev invariants. Furthermore, the Jones polynomial not only introduced a sophisticated method of analyzing knots in 3-manifolds but also related the mathematical theory of knots to other disciplines of mathematics and theoretical physics, for instance, 15

See [Prz32] for Przytycki and Traczyk motivation to work on a generalization of the Jones polynomial.

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2 History of Knot Theory from Gauss to Jones

statistical mechanics, quantum field theory, operator algebra, graph theory, and computational complexity. While it is a simple and effective tool to recognize knots, the Jones polynomial is also of great use to biologists and chemists, specifically for the analysis of DNA [SCKSSWW]. Much of the contemporary investigations in knot theory are somehow connected to the construction by Vaughan Jones and involve the notion of categorification. The inclusion of homology theories in topological contexts generates beautiful and very productive mathematical structures, opening a new world of possibilities. In particular, the Khovanov homology, Yang-Baxter homology, and categorification of skein modules will be discussed in the book.

Lecture 3 From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

This lecture is divided into two parts. The first part of the lecture is more elementary and explores the Fox 3-coloring invariant and its generalization into quandle invariants. Here we also describe the Wirtinger’s and Dehn’s presentations of the fundamental group of the link complement and their relation to Fox colorings. The second part of the lecture describes how the idea of Fox colorings can develop into the sophisticated notion of Yang-Baxter operator and its homology. The Yang-Baxter equation was extensively studied by Chen N. Yang in 1968 and by Rodney J. Baxter in 1971. The relation between the equation and knot theory was first noted by Vaughan F. R. Jones, when he constructed the Yang-Baxter operator yielding his polynomial and the HOMFLYPT polynomial. Yang-Baxter homology in full generality was introduced by Victoria Lebed and the first author in 2012. Since then, it has become of substantial interest to knot theorists.

3.1 Part I: Fox Coloring, Wirtinger’s Presentation, and Dehn’s Presentation 3.1.1 Introduction to Link Invariants Classical knot theory studies the embeddings of a circle (knot), or several circles (link), up to natural deformations in .R3 (or in .S3 = R3 ∪ ∞) emphasizing that one of the fundamental problems is the classification. This classification, or enumeration, is done up to the natural movement in space which is called an ambient isotopy. Recall that in 1927, Reidemeister showed that two link diagrams (possibly oriented) are isotopic if and only if they are connected by a finite sequence of moves, called

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_3

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3 From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

Reidemeister moves,1 and planar isotopy. Figure 3.1 shows the three Reidemeister moves denoted by . R1 , . R2 , and . R3 . Following this idea, to distinguish knots and links, we look for invariants of links, that is, properties of links that remain unchanged under ambient isotopy. In other words, to show that a property .P(D) of a diagram D is an invariant, one has to check that it is preserved under Reidemeister moves.

Fig. 3.1: Reidemeister moves . R1 , . R2 , and . R3 When looking for invariants of links, some criteria must be taken into account. For instance, how easy is the invariant to compute and how good is it at distinguishing links? Simple examples of invariants include the number of components of a link and the linking number defined in 1833 by Carl F. Gauss using a certain double integral [Gau3]; see Lecture 1.

3.1.2 Fox 3-Colorings and Generalization to n-Colorings The Fox 3-coloring invariant was developed by the famous American topologist Ralph H. Fox (1913–1973). In 1956, while on sabbatical from Princeton University, Fox went to teach at Haverford College, and he used the idea to explain knot theory to undergraduate students in an attempt to make the subject accessible to everyone. Fox, together with his student Richard H. Crowell (1928–2006), wrote a book that became very popular (e.g., it has Japanese and Russian translations) on knot theory [CF]. However, the book only mentions Fox 3-colorings in the exercises (Chapter VI, Exercises 6–7; Chapter VIII, Exercises 8–10). In the 1970s, Louis H. Kauffman and José M. Montesinos (born in November 1944) extensively used and popularized this invariant. Definition 3.1.1 A link diagram D is said to have a Fox 3-coloring if the arcs of the diagram can be colored in such a way that at any crossing either all three colors appear or only one color appears; see Fig. 3.2 where the arcs of the diagram are denoted by a, b, and c. The number of different Fox 3-colorings of the link diagram D is denoted by .tri(D). A Fox 3-coloring using only one color is said to be a trivial Fox 3-coloring. Here, we consider the arcs of the diagram in the literal sense, that is, from undercrossing to undercrossing.2 Figure 3.2 shows a projection of the right-trefoil 1 These moves were first envisioned by the Scottish scientist James Maxwell around 1870 and proved independently by Alexander and Briggs in 1927.

3.1 Part I: Fox Coloring, Wirtinger’s Presentation, and Dehn’s Presentation

39

knot with its arcs a, b, and c colored red, blue, and green, respectively. The proof that .tri(D) is unaffected under Reidemeister moves requires checking a number of cases, all of them straightforward. We leave the proof of the following lemma as an exercise.

Fig. 3.2: Coloring arcs of a link diagram D

Lemma 3.1.2 The number of Fox 3-colorings is an ambient isotopy link invariant. Let .O denote the unknot. Certainly, .tri(O) = 3. As shown in Fig. 3.2, the trefoil knot allows a nontrivial Fox 3-coloring. In fact, ( ) .= 9. Thus, the trefoil knot and the unknot are different. More generally, from Lemma 3.1.2, it follows that if a knot K has a nontrivial Fox 3-coloring, then .tri(K) > 3 which implies that the knot is nontrivial.

The following lemma discusses some basic properties of Fox 3-colorability. For a connection between the number of Fox 3-colorings and the Jones polynomial, see Lecture 5. For a deeper discussion on this invariant, the reader is referred to [Prz13]. Lemma 3.1.3 The Fox 3-coloring invariant satisfies the following properties: 1. .tri(L) is always a power of 3. 2. .tri(L1 )tri(L2 ) = 3tri(L1 #L2 ), where .# denotes the connected sum of links; see Fig. 3.3. The notion of connected sum in knot theory refers to the intuitive idea of tying one knot and then another using the same cord. More precisely, to obtain . L1 #L2 , the links are supposed to be in different copies of .S 3 , then a “small arc” is removed from an unknotted part of the diagrams, and finally the resulting cords are identified as depicted in Fig. 3.3; compare with Definition 4.6.11. Proof Hint: As in Fig. 3.2, denote the colors red, blue, and green by a, b, and c, respectively. Treat them as elements of the field .Z3 . Notice that the condition for Fox 3-coloring can be characterized to satisfy the congruence .a + b + c ≡ 0 mod 3 (i.e., .a + b + c = 3 in .Z3 ), where a, b, and c are colors associated with a given crossing. Observe that the Fox 3-colorings, . f : arcs −→ {0, 1, 2}, form a linear space over .Z3 . 2 A component with no crossings is considered to be an arc.

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3 From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

Fig. 3.3: Connected sum of the right-trefoil .31 knot with its mirror image, the left-trefoil knot .31 , also called the square knot

Crowell, who gave talks about Fox 3-colorings to teachers, started to think on possible ways of generalizing the concept to n-colorings. In an article for teachers in 1961, Crowell writes that it was Hale F. Trotter (1931–2022), a professor from Princeton, who gave him the idea of the generalization. Trotter is particularly wellknown in knot theory because he proved the existence of oriented knots that are different from the same knot when given a different orientation. In other words, by deformations of the space, one orientation cannot be changed to the other. The gist of the generalization of Fox’s idea arises from the fact that in 3-colorings, we work with the set of “colors” .Z3 . Additionally, at any crossing v of a link diagram D, as in Fig. 3.2, the property .a + b + c ≡ 0 mod 3 is satisfied. This property can be extended to .Zn as .2b ≡ a + c mod n (twice the color of the bridge equals the sum of the color of the tunnels). The following definition formalizes the result and sets our notation. Definition 3.1.4 A Fox n-coloring of a diagram D is a function . f : arcs → Zn , satisfying the property that every arc is “colored” by an element of .Zn = {0, 1, 2, 3, . . . , n − 1} in such a way that at each crossing the sum of the colors of the undercrossings is equal to twice the color of the overcrossing modulo n. That is, if at a crossing v the overcrossing is colored b, and the undercrossings are colored by a and c, then .2b − a − c ≡ 0 mod n; see Fig. 3.2. The set of Fox n-colorings of a diagram D is denoted by .Coln (D), and the number of Fox n-colorings is denoted by .coln (D). Notice that for .n = 3 the congruence .2b − a − c ≡ 0 mod n is the same as a + b + c ≡ 0 mod 3. We will prove that the number of Fox n-colorings is an invariant of links. That is, we verify that for a link diagram D, .coln (D) is preserved by the Reidemeister moves. To this end, the interpretation “twice the color of the bridge equals the sum of the color of the tunnels” is very practical.

.

Fig. 3.4: Invariance of .coln (D) under Reidemeister moves . R1 and . R2

3.1 Part I: Fox Coloring, Wirtinger’s Presentation, and Dehn’s Presentation

41

From Fig. 3.4, we conclude that .coln (R1 (D)) = coln (D) and .coln (R2 (D)) = coln (D). Consider now a braid-like Reidemeister move . R3 as in Fig. 3.5.

Fig. 3.5: Invariance of .coln (D) under Reidemeister move . R3 Then, .coln (R3 (D)) = coln (D) which completes the proof that .coln (D) is a link invariant. In the following example, the process of how to find .coln (D) for a diagram D is illustrated. Example 3.1.5 We find (

), the number of 5-colorings of the figure-eight knot.

We start by coloring two arcs with any “colors” a and b at the top of the diagram. Making our way to the bottom, we deduce the other colors following the rule of n-colorings.

Fig. 3.6: Calculating the number of Fox 5-colorings of the figure-eight knot From Fig. 3.6, we need .3a − 2b ≡ 3b − 2a mod n and .a ≡ 5b − 4a mod n. It follows from both equations that .5(a − b) ≡ 0 mod n, which holds for .n = 5. We ) .= 25. Finally, conclude that there are f ive choices for each a and b and thus ( instead of looking for modular equalities from the diagram, we could consider the group .{a , b | 5(a − b) = 0} = Z ⊕ Z5 . This abelian group is called the universal or fundamental group of the Fox coloring of the link diagram D. In general, for a link diagram D, the universal group .Col(D) is defined as in Fig. 3.7, where .arcs(D) denotes the set of all arcs of D. The generators of the group are indexed by the arcs and the relations are given by the crossings. For a discussion on the relation of this universal group and the first homology of the double branch cover of .S 3 with a link as the branch set, see [Prz13].

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3 From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

Fig. 3.7: Presentation of the group .Col(D)

The following properties of Fox n-colorability are similar to those of tricolorability. Lemma 3.1.6 1. Reidemeister moves preserve the number of n-colorings, .coln (D); thus, it is a link invariant. 2. .coln (L1 )coln (L2 ) = n(coln (L1 #L2 )). Exercise 3.1.7 (1) Show that the . Zn -module .Coln (D) is preserved by Reidemeister moves. (2) Find .Col(L1 #L2 ) when .Col(L1 ) and .Col(L2 ) are known. In 1998, Louis H. Kauffman and Frank Harary formulated the following conjecture [HaKa]: Conjecture 3.1.8 Let D be a reduced alternating diagram of a knot K having determinant p, where p is prime. Then every nontrivial p-coloring of D assigns different colors to different arcs. The conjecture was proven by Thomas W. Mattman and Pablo Solis in [MaSo]. The generalization of this conjecture was formulated in [APS2] in 2003. Conjecture 3.1.9 Let D be a reduced alternating diagram of an alternating prime link. Let .Z |arcs| denote the free abelian group .Z |arcs| = {arcs(D) | ∅}. Consider the β

→ Col(D). Then . β is injective on the arcs of D, that is, . β(ai ) = β(a j ) map .Z |arcs| − for .i = j. This conjecture was proven in [BGMMP]. Definition 3.1.10 The bridge number of a diagram is the number of arcs of the diagram which either have only overcrossings or are closed components. The bridge number of the link L is the minimal bridge number of diagrams representing L. The following exercise connects Fox colorings with the bridge number of a link.

3.1 Part I: Fox Coloring, Wirtinger’s Presentation, and Dehn’s Presentation

43

Exercise 3.1.11 Show that the minimal number of generators of .Col(L) is bounded from above by the bridge number of L. Hint: Notice that if a link has bridge number n, then it has a diagram with n maxima and n minima.

3.1.3 Wirtinger’s and Dehn’s Presentations of the Knot Group Wilhelm Wirtinger (1865–1945), an Austrian mathematician, outlined a method of finding a knot group presentation in [Wir] at a lecture delivered at a meeting of the German Mathematical Society in 1905 in Meran. This method is now wellknown as Wirtinger’s presentation. While the proof that the presentation describes the fundamental group of the knot complement requires the Van Kampen-Seifert theorem, proving that it is a link invariant is much more basic and only requires that the Reidemeister moves hold. In 1910, Max Dehn in [Deh2] introduced a presentation of the knot group that is different from Wirtinger’s while constructing examples of “Poincaré spaces” and proving that a knot is unknotted if and only if the knot group is abelian. In this section, we will give a detailed description of both presentations.

(a) Positive crossing

(b) Negative crossing

Fig. 3.8: The sign convention of oriented crossings

Definition 3.1.12 (Wirtinger’s Presentation) Consider an oriented diagram D. The sign convention of positive and negative crossings is illustrated in Fig. 3.8. The Wirtinger group of the diagram has the following presentation with generators of the group indexed by arcs of the diagram (if the component has no crossings, we consider this to be an arc) and relations indexed by the crossings. Let . x1, . . . , xn be generators corresponding to the arcs and .r1, . . . , rm denote relations corresponding to the crossings of D with the convention given in Fig. 3.9. Thus, WD = {x1, . . . , xn | r1, . . . , rm }.

.

Theorem 3.1.13 .WD is a link invariant.

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3 From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

The theorem can be proved by showing that .WD is invariant under the Reidemeister moves. Theorem 3.1.14 ([Wir]) .WD is isomorphic to the fundamental group of the link complement.

+1

(a) Relation

:

−1 −1 .

+1

+1

(b) Relation

:

+1

−1 −1 .

Fig. 3.9: The two possible relations from the arrows placed under each arc in an oriented crossing in such a way that the crossing formed between the arc and arrow forms is a positive crossing

Exercise 3.1.15 Show that the number of arcs in the diagram of a link is equal to the crossing number plus the number of trivial components of the diagram. Exercise 3.1.16 Prove that for a knot diagram, one of the relations .{r1, . . . , rm } in Wirtinger’s presentation may be omitted. What about a link diagram? Example 3.1.17 Consider an oriented right-handed trefoil with small arrows drawn according to Wirtinger’s algorithm as shown in Fig. 3.10. From the three crossings, we obtain the following relations: .

ca = bcbc = abca = ab.

Therefore, Wirtinger’s presentation is .

Exercise 3.1.18 Consider the relations in Example 3.1.17, and show that you can write one of the relations from the other two. In particular, show that

Exercise 3.1.19 Show that the group .{a, b, |aba = bab} is non-abelian; thus, the is nontrivial. trefoil

3.1 Part I: Fox Coloring, Wirtinger’s Presentation, and Dehn’s Presentation

45

Fig. 3.10: An oriented trefoil knot with small arrows drawn according to Wirtinger’s presentation convention

Exercise 3.1.20 Show that the map f

.

{a, b, |aba = bab} → − S3

given by . f (a) = s1, f (b) = s2 , where .S3 = {s1, s2 |s12 = s22 = 1, s1 s2 s1 = s2 s1 s2 }, is an epimorphism. Definition 3.1.21 (Dehn’s Presentation) Let K be a link diagram, label each enclosed region of the diagram, and consider the outer region to be the infinite region. Relations from the crossing are given by loops starting from the infinite region and then going around a crossing from over an undercrossing and under an overcrossing. Each time the loop intersects a region, say region B from above to below, we write B, but if the loop intersects a region, say C, from below to above, then we write .C −1 as shown in Fig. 3.11. Following Dehn’s convention, we may present the loops from Wirtinger’s presentation in terms of Dehn’s presentation as follows: For Fig. 3.11, we have . xi+1 = AB−1, xk = BC −1, xi = DC −1, xk = AD−1 . For Fig. 3.11, we have . xi+1 = AB−1, xk = CB−1, xi = DC −1, xk = DA−1 . Corollary 3.1.22 The group formed by Dehn’s presentation describes the fundamental group of the link complement.

Example 3.1.23 Consider the diagram of an oriented right-handed trefoil as shown in Fig. 3.12. According to Dehn’s convention, from each crossing, we have the following relations: −1 −1 .r1 : BD CE r2 : AD−1 BE −1 r3 : CD−1 AE −1 . Wirtinger’s generators, as shown in Fig. 3.12, written in terms of Dehn’s presentations are as follows: −1 .a = AE , b = BE −1, c = CE −1 . A loop from the infinite region and then back without intersecting any other regions produces a trivial loop; therefore, .E = 1. We can reduce the relations of the crossing to the following:

46

3 From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

+1

−1

(a) Relation

−1 .

+1

(b) Positive crossing.

(c) Negative crossing.

Fig. 3.11: Each region of the diagram is labeled and to find the relation from each crossing, we take a loop placed above the under-arc and below the over-arc of the crossing r1 : BD−1 C = 1,

.

r2 : AD−1 B = 1,

r3 : CD−1 A.

From .r1 and .r2 , we have .C = BAB−1 , and from .r2 and .r3 , we have .C = A−1 BA. Hence, we obtain the braid relation . BAB = ABA and .

π1 (R3 \31 ) = {A, B|BAB = ABA}.

=

=

= Fig. 3.12: An oriented trefoil knot with its faces labeled according to Dehn’s presentation convention. Notice that . D = ac = AC = ba = BA = cb = CB

3.1.4 Constructing Invariants from Magmas: Quandles The construction of knot invariants from diagrams can be generalized from Fox colorings in the following way. A magma is a pair .(X, ∗), where X is any set and .∗ : X × X −→ X is a binary operation on X. We can naively look for invariants of links coloring the arcs of the diagram by elements of X. Since we would like to consider oriented link diagrams, we have positive and negative crossings; therefore, we need two binary operations .∗ : X × X −→ X and .∗ : X × X −→ X. We call X the set of colors and the operations

3.1 Part I: Fox Coloring, Wirtinger’s Presentation, and Dehn’s Presentation

47

∗ and .∗ give us the rules of coloring every crossing. Let v be any crossing of an oriented link diagram D and suppose that for all v, positive or negative, the condition from Fig. 3.13 is satisfied. Define the colorings of arcs of the diagram D by elements of X, as functions .ϕ : arcs(D) −→ X satisfying the above conditions. For finite X, denote the number of colorings of the diagram D as .colX (D). Notice that the Fox 3-coloring is the special case where X consists of three colors and the operation is given by .a ∗ b = a ∗ b = 2b − a.

.

The binary operations .∗ and .∗ can be interpreted as an “overcrossing acting on an undercrossing.” In order for .colX (D) to be an invariant, it must be preserved by the Reidemeister moves. In a similar way as we did for Fox n-colorings, we study the effect of . R1 , . R2 , and . R3 on the operations .∗ and .∗.

Fig. 3.13: Binary operations on positive (left) and negative (right) crossing

Figure 3.14 illustrates the effect of Reidemeister type I and II moves. Observe that in order for .colX (D) to be unaffected by . R1 , we need the equalities .a ∗ a = a and .a ∗ a = a. Additionally, for .col X (D) to be invariant under . R2 , we need .(a ∗ b) ∗ b = a and .(b ∗ a) ∗ a = b. The operations .∗ and .∗ satisfying the last two equalities are said to be inverse.

Fig. 3.14: Effect of Reidemeister moves

Finally, the effect of a type III Reidemeister move is shown in Fig. 3.15, resulting in the need for right self-distributivity, i.e., .(a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c).

48

3 From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

It is important to remark that when considering orientation in verifying Reidemeister move invariance, as in Figs. 3.14 and 3.15, there are two possible orientations. However, due to the well-known theorem by James W. Alexander (1888–1971), we know that every link can be presented as a closed braid [Ale2]. Furthermore, we can invoke Andrey A. Markov’s (1903–1979) theorem from 1936 from which it follows that it is enough to consider the third Reidemeister move with all positive crossings, the second Reidemeister move with parallel strings, and the first Reidemeister move by always keeping the braid structure. The reader is referred to Lecture 8 for a further discussion of this result. Therefore, if we want .colX (D) to be a link invariant, we need .(X, ∗) to be an algebraic structure satisfying certain conditions, which David Joyce called in his 1979 PhD thesis a quandle [Joy].

Fig. 3.15: Third Reidemeister move leads to right self-distributivity

Definition 3.1.24 Let .(X, ∗) be a magma. Then: 1. If .∗ is right self-distributive, i.e., .(a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c), then .(X, ∗) is called a shelf. 2. If a shelf .(X, ∗) satisfies the idempotency property, i.e., .a ∗ a = a for any .a ∈ X, then it is called a right spindle or just a spindle. 3. If for a shelf .(X, ∗) there exists an inverse operation .∗, then it is called a rack. 4. A rack with idempotency is called a quandle.

3.2 Part II: The Yang-Baxter Operator and Its Homology

49

3.2 Part II: The Yang-Baxter Operator and Its Homology 3.2.1 History The celebrated Yang-Baxter equation was first developed in 1968 by the Chinese theoretical physicist Chen N. Yang (born in October 1922). Then in 1971, the equation appeared in the work on statistical mechanics by the Australian physicist Rodney J. Baxter (born in February 1940 in London). In 1948, Yang got his PhD from the University of Chicago under the supervision of the Hungarian-American physicist Edward Teller (1908–2003). He was awarded the Nobel Prize in Physics in 1957, together with Tsung-Dao Lee (born November 1926), for his theory on parity violation. On the other hand, Baxter received his PhD in 1964 from the Australian National University in Canberra. His work focuses mainly on exactly solved models such as the six-vertex model for which the Yang-Baxter equation plays an important role. The Yang-Baxter equation was formerly known as the star-triangle equation3 and has specially become attractive to topologists given its application to knot theory. The relation between the equation and knot theory was first discovered by Vaughan F. R. Jones who won the Fields Medal in 1990. In this section, we show how the construction of certain knot invariants can lead to the Yang-Baxter equation, and the ground is set for future discussions of homology theories motivated by this concept.

3.2.2 The Set-Theoretic Yang-Baxter Equation It is possible to consider more general colorings than racks and quandles when two parts of an over-strand are allowed to have different colors. As previously done, let X be a finite set of colors and let D be an oriented link diagram. Define the colorings of semiarcs4 of the diagram D, by elements of X, as functions .ϕ : semiarcs(D) −→ X. To define allowed colorings at a crossing v, we introduce the maps . R, R : X × X −→ X × X for which we use the notation . R(a, b) = (R1 (a, b), R2 (a, b))5 and . R(a, b) = (R1 (a, b), R2 (a, b)). Allowed colorings R are shown in Fig. 3.16, where (a, b) −−−−−−−→ (R1 (a, b), R2 (a, b)) if the crossing v is R positive, while .(a, b) −−−−−−−→ (R1 (a, b), R2 (a, b)) if the crossing v is negative. .

For convenience, instead of working with the coordinates . R1 (a, b), . R2 (a, b), R1 (a, b), and . R2 (a, b), we work with the functions R and . R.

3 The name Yang-Baxter equation was coined by Ludvig Faddeev. 4 Here we understand a semiarc to be the piece of string from crossing to crossing, independently of being an over-strand or an under-strand. 5 From the context, it should be clear if we are referring to the Reidemeister moves or to the map . R : X × X −→ X × X.

50

3 From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

Fig. 3.16: General semiarc colorings

From Fig. 3.17, we observe that in order for .colX (D) to be unaffected by the second Reidemeister move, one needs . RR = Id = RR. That is, R is invertible with inverse . R. On the other hand, in order for .colX (D) to be unaffected by the third Reidemeister move, we need R to satisfy the equation: .(R × Id)(Id × R)(R × Id) = (Id × R)(R × Id)(Id × R), called the set-theoretic Yang-Baxter equation (a term coined by Vladimir Drinfeld6 in 1992); see Fig. 3.17 [PW1].

Fig. 3.17: Set-theoretic Yang-Baxter equation If R satisfies this equation, then R is called a set-theoretic pre-Yang-Baxter operator. If R is additionally invertible, then it is said to be a set-theoretic YangBaxter operator. Exercise 3.2.1 Show that .colX (D) can be written as the following state sum formula: colX (D) =

E

| |

.

w(v, ϕ),

ϕ ∈ col X (D) v ∈ cr(D)

where .cr(D) is the set of all crossings of D and .w(v, ϕ) is either 1 or 0, depending on whether the coloring is allowed or not. This state sum formula connects us with statistical mechanics [HJ, Kau7, Prz13].

6 Born in 1954 in Kharkiv; he won the Fields Medal in 1990.

3.2 Part II: The Yang-Baxter Operator and Its Homology

51

Fig. 3.18: Boltzmann weights for positive and negative crossings

3.2.3 From Yang-Baxter Operators to Knot Theory In quandle and set-theoretic Yang-Baxter colorings of an oriented link diagram, it is assumed that at every crossing, a coloring of the input semiarcs defines a unique coloring of the output semiarcs. However, this condition can be relaxed in a natural way by allowing any coloring and then associating, with any crossing, a weight from a fixed commutative ring. Notice that for set-theoretic Yang-Baxter operator, this weight is either 1 or 0 depending on whether the coloring is allowed or not, respectively. Let X be a fixed finite set of colors, .k a fixed commutative ring, and D an oriented link diagram. Assign a color from X to semiarcs of D allowing different weights from .k for every crossing v. In other words, we are considering functions . ϕ : semiarcs(D) −→ X. Weights associating to crossings are called Boltzmann weights. The term is borrowed from statistical mechanics and is often used in knot theory when dealing with state sums; see, for example, [CKS2]. Depending on whether the crossing v is positive or negative, it is assigned the weight . Rca db ∈ k or .

ab d

Rc

∈ k, respectively, as depicted in Fig. 3.18.

Using Boltzmann weights, we can generalize the number of colorings, .col(X,BW ) (D), to a state sum by multiplying the weights over all crossings v and adding over all colorings .ϕ [Jon7, Tur2], as:

.

col(X,BW ) (D) =

E

| |

.

Rˆca db (v),

ϕ ∈ col X (D) v ∈ cr(D) ab

where .cr(D) denotes the set of all crossings of D and . Rˆca db (v) is either . Rca db or . Rc d , depending on whether v is a positive or negative crossing, respectively. It is natural now to check the invariance of the state sum under Reidemeister moves. Checking the invariance using the “coordinate” notation from Fig. 3.18 would be too complicated. Instead, we can interpret this notation in a more familiar way using linear algebra as follows. Pairs of colors .(a, b) can be seen as the basis of a linear space (or module) in such a way that we are working with the entries of the matrix given by the map R defined as follows. Let .V = kX be a linear space (or

52

3 From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

module), and then the given Boltzmann weights . Rca db can be used to define the linear map . R : V ⊗ V −→ V ⊗ V. This map can be naively interpreted as “pairs of colors .(a, b) are sent to linear combinations of color pairs.” A naive physical interpretation is that particles a and b collide and scatter as linear combination of other particles. That is, the map R is defined on the basis . X × X of .V ⊗ V as: .

R(a, b) =

E

Rca db · (c, d).

(c,d)

Figure 3.19 shows that we need R and . R to be inverses of each other, i.e., RR = Id = RR, where . R : V ⊗ V −→ V ⊗ V.

.

Fig. 3.19: Invariance of .col(X,BW ) (D) under the second Reidemeister move Recall that to check invariance under the third Reidemeister move, it is sufficient to consider the case where all crossings are positive. From Fig. 3.20, we observe that in order to obtain invariance of .col(X,BW ) (D) under the third Reidemeister move, we need the following equality, called the Yang-Baxter equation: (R ⊗ Id)(Id ⊗ R)(R ⊗ Id) = (Id ⊗ R)(R ⊗ Id)(Id ⊗ R),

.

for a map .V ⊗ V ⊗ V −→ V ⊗ V ⊗ V. As we previously mentioned, the map R : V ⊗ V −→ V ⊗ V is said to be a Yang-Baxter operator if it is invertible and it satisfies the Yang-Baxter equation. In the case where R is not necessarily invertible, it is called a pre-Yang-Baxter operator, which is enough to define Yang-Baxter homology.

.

Fig. 3.20: Invariance of .col(X,BW ) (D) under the third Reidemeister move

3.2 Part II: The Yang-Baxter Operator and Its Homology

53

Finally, we remark that the first Reidemeister move requires a delicate balancing of the state sum formula, and the reader is referred to [Tur2] for a detailed description.

Example 3.2.2 We conclude our introductory discussion of the Yang-Baxter ideas with the following example. Let V be 2-dimensional, and consider the canonical basis given by .e0 and .e1 , so that .V = span(e0, e1 ). Then, the simplest Yang-Baxter operator, .V ⊗ V −→ V ⊗ V, can be represented with the following matrix: R .0 ⊗ 0 .0 ⊗ 1 .1 ⊗ 0 .1 ⊗ 1 0⊗0 1 0 0 0 .0 ⊗ 1 0 0 1 0 .1 ⊗ 0 0 1 0 0 .1 ⊗ 1 0 0 0 1 .

R .0 ⊗ 0 .0 ⊗ 1 .1 ⊗ 0 .1 ⊗ 1 0⊗0 1 0 0 0 deforms to .0 ⊗ 1 0 0 1 0 .− −−−−−−→ 0 1 0 0 .1 ⊗ 0 .1 ⊗ 1 0 0 0 1 .

⎡q 0 ⎢ ⎢0 q − q−1 .⎢ ⎢0 1 ⎢ ⎢0 0 ⎣

0 1 0 0

0⎤⎥ 0⎥⎥ 0⎥⎥ q⎥⎦

In this case, we have that . R is given by the matrix: ⎡q−1 ⎢ ⎢ 0 .⎢ ⎢ 0 ⎢ ⎢ 0 ⎣

0 0 0 1 1 q−1 − q 0 0

⎡1 0 0 0⎤ 0 ⎤⎥ ⎢ ⎥ ⎢0 1 0 0⎥ ⎥ 0 ⎥ which implies ⎥ .− − − − − − − − − − → . RR= . ⎢ ⎢0 0 1 0⎥ . 0 ⎥⎥ ⎢ ⎥ ⎢0 0 0 1⎥ q−1 ⎥⎦ ⎣ ⎦

Now, in determining the minimal polynomial for the matrix . RR, notice that: ⎡q − q−1 0 0 0 ⎤⎥ ⎢ −1 ⎢ ⎥ 0 0 0 q − q −1 ⎥ = .(q − q−1 )Id, .R − R = . ⎢ −1 ⎢ 0 0 ⎥⎥ 0 q−q ⎢ ⎢ 0 0 0 q − q−1 ⎥⎦ ⎣ and from this, we get that . R2 − (q − q−1 )R − Id = 0. Therefore, the minimal polynomial is equal to .λ2 − (q − q−1 )λ − 1. The above equation leads to the Jones skein relation after appropriate balancing; see [Jon7, Tur2]. Thus, if we make the matrix R column unital (e.g., stochastic), we obtain:

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3 From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

⎡1 0 ⎢ ⎢ ⎢ −1 ⎢ ⎢0 q − q ⎢ 1 + q − q−1 ⎢ mod = . ⎢ .R ⎢ ⎢ 1 ⎢0 ⎢ 1 + q − q−1 ⎢ ⎢ ⎢ ⎢0 0 ⎣

0 0⎤⎥ ⎥ ⎥ ⎥ 1 0⎥⎥ ⎡⎢1 0 ⎥ ⎢0 1 − y 2 ⎥=⎢ ⎥ ⎢0 y 2 ⎥ ⎢ 0 0⎥⎥ ⎢⎣0 0 ⎥ ⎥ ⎥ 0 1⎥⎦

0 0⎤⎥ 1 0⎥⎥ 0 0⎥⎥ 0 1⎥⎦

1 q − q−1 2 , then .1 − y = . It can be checked that 1 + q − q−1 1 + q − q−1 this is a Yang-Baxter operator; see [PW1, Wan]. A generalization of this idea will be discussed later.

and by defining . y 2 =

3.2.4 Homology of the Yang-Baxter Operator The notion of homology groups was introduced in 1895 by Jules Henri Poincaré (1854–1912) in his Analisis Situs paper [Poi1] to study properties of manifolds; see Lecture 2 for a historical remark on the relation between algebraic topology and knot theory. We now recall the definition of a chain complex and its homology. Definition 3.2.3 Consider the following descendant sequence of abelian groups or modules .Cn connected by the homomorphisms .∂n : Cn −→ Cn−1 , called boundary operators: ∂n+3

.

∂n+2

∂n+1

∂n

∂n−1

· · · −−−→ Cn+2 −−−→ Cn+1 −−−→ Cn −−→ Cn−1 −−−→ · · ·

The system .(Cn, ∂n ) satisfying .∂n−1 ◦ ∂n = 0 is called a chain complex. The previous condition is equivalent to requiring that .im(∂n+1 ) ⊂ ker(∂n ), where .im(∂n+1 ) denotes the image of the boundary map and .ker(∂n ) denotes its kernel. We define the nth homology group as the quotient: Hn :=

.

ker(∂n ) . im(∂n+1 )

The elements of .Hn are called homology classes. Elements of .ker(∂n ) are called ncycles and are denoted by . Zn = ker(∂n ); elements of .im(∂n+1 ) are called n-boundaries and are denoted by . Bn = im(∂n+1 ). In particular, if the equality .im(∂n+1 ) = ker(∂n ) holds for all n, then the chain complex .(Cn, ∂n ) is said to be exact. The homology theory of associative structures, such as groups and rings, has been extensively studied beginning with the work of Heinz Hopf (1894–1971), Witold

3.2 Part II: The Yang-Baxter Operator and Its Homology

55

Hurewicz (1904–1956), Samuel Eilenberg (1913–1998), and Gerhard P. Hochschild (1915–2010). On the other hand, the homology of nonassociative distributive structures, such as quandles, was, in a way, undervalued until recently. Distributive structures have been studied for a long time; we remark that in 1880 Charles S. Peirce (1839–1914) emphasized the importance of right self-distributivity in algebraic structures. However, the first homology theory related to a self-distributive structure was constructed in the early 1990s by Roger Fenn, Colin Rourke, and Brian Sanderson. They introduced the homology theory of racks motivated by higher-dimensional knot theory in [FRS]. In 1998, J. Scott Carter, Seiichi Kamada, and Masahico Saito refined the ideas to define the homology of quandles [CKS1, CKS2]; See also [CJKLS, CJKS]. Then in 2004, Carter, Saito, and Mohamed Elhamdadi generalized this homology by defining a (co)homology theory for set-theoretic Yang-Baxter operators, and they also constructed cocycle invariants [CES]. The homology theory of general Yang-Baxter operators was independently developed by Victoria Lebed [Leb] and the first author [Prz26]. This homology theory is equivalent to that defined by Carter, Kamada, and Saito when restricted to set-theoretic Yang-Baxter operators; see [PW1]. To define Yang-Baxter homology, it is convenient to have the terminology of presimplicial7 and precubic sets and modules. These concepts take into account the fact that, in most homology theories, the boundary map .∂n : Cn −→ Cn−1 can be decomposed as an alternating sum of other maps. The formal definitions mostly follow [Lod]; see also [Prz23]. Definition 3.2.4 Let . Xn with .n ≥ 0 be a sequence of sets and consider the maps di := di,n : Xn −→ Xn−1 , .0 ≤ i ≤ n, called face maps or face operators, such that:

.

.

di d j = d j−1 di

f or any i < j.

The system .(Xn, di ) satisfying the above equality is called a presimplicial set. Analogously, if .Cn with .n ≥ 0 is a sequence of .k-modules, for a fixed commutative ring .k (e.g., .Cn = kXn ), and the maps .di := di,n : Cn −→ Cn−1 , .0 ≤ i ≤ n, are homomorphisms satisfying: .

di d j = d j−1 di

f or any i < j,

then the system .(Cn , di ) is called a presimplicial module. If .(Cn , di ) is a presimplicial module, then the system .(Cn , ∂n ) is a chain complex for: .

∂n =

n E

(−1)i di .

i=0 ε : X −→ Definition 3.2.5 Let . Xn with .n ≥ 0 be a sequence of sets and let .diε := di,n n Xn−1 be maps called face maps or face operators, where .ε ∈ {0, 1}, .0 ≤ i ≤ n. If the condition

7 The concept was introduced in 1950 by Samuel Eilenberg (1913–1998) and Joseph A. Zilber (1923–2009) under the name semi-simplicial complex [EZ].

56

3 From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology .

δ diε d jδ = d j−1 diε

f or i < j

is satisfied, then the system .(Xn, diε ) is called a precubic set. If .Cn = kXn , then the system .(Cn , ∂n ) is a chain complex for: .

∂n =

n E (−1)i (di0 − di1 ). i=1

The notion of a precubic module is defined analogously to the presimplicial module in Definition 3.2.4. We are now ready to formally state the definition of the Yang-Baxter homology following [PW1]. Definition 3.2.6 Let .k be a commutative ring and let .V = kX be the free .k-module with basis X. Moreover, let . R : V ⊗ V −→ V ⊗ V be a column unital Yang-Baxter operator. We define the precubic module .(Cn, diε ) by setting .Cn (R) = V ⊗n where the l and . d r are defined as in Fig. 3.21.8 As illustrated, we work from the face maps .di,n i,n top to the bottom, at each crossing we apply the Yang-Baxter operator R, and at the l or . d r , respectively. end we delete the first tensor factor or the last tensor factor of .di,n i,n Following Definition 3.2.5, we define the Yang-Baxter chain complex .(Cn, ∂n ) with: n E l r . ∂n = (−1)i (di,n − di,n ), i=1

and finally, the homology of the Yang-Baxter operator R is defined as: .

Hn (R) =

ker(∂n ) . im(∂n+1 )

To conclude the discussion, we describe the first nontrivial calculation of the homology of Yang-Baxter operators; for details the reader is referred to [PW2]. As before, let .k be a commutative ring and let .V = kX be a free .k-module over the basis . X = {v1, v2, . . . , vm } with the ordering .v a ≤ vb if and only if .a ≤ b. Recall that a .k-linear map . R : V ⊗ V −→ V ⊗ V is a Yang-Baxter operator if it satisfies the YangBaxter equation and it is invertible. Jones discovered that the Yang-Baxter operator on level m given by the formula below leads to the HOMFLYPT polynomial:

.

Rca db

⎧ q, ⎪ ⎪ ⎪ ⎪ ⎨ 1, ⎪ = ⎪ q − q−1, ⎪ ⎪ ⎪ ⎪ 0, ⎩

if a = b = c = d; if d = a = b = c; if c = a < b = d; otherwise.

8 For convenience we use the notation .di,r n and .di,l n instead of .di0 and .di1 , respectively.

3.2 Part II: The Yang-Baxter Operator and Its Homology

57

Fig. 3.21: Graphical interpretation of the face maps .dil and .dir

Xiao Wang and the first author in [PW2] adjusted the matrix above to be a column unital matrix and showed that for each m, . R(m) is also a Yang-Baxter operator. Precisely, the matrix is obtained by adding the elements in each column and dividing every element of the column by this sum, as done in Example 3.2.2. In the next 1 . theorem, as in Example 3.2.2, we set . y 2 = 1 + q − q−1 Theorem 3.2.7 (PW) Let .k = Z[y ±1 ], m be a positive integer, and .Vm be the free .k-module generated by the set . Xm = {v1, v2, . . . , vm } with the ordering .v a ≤ vb if and only if .a ≤ b. Then the .k-linear operator . R(m) : Vm ⊗ Vm −→ Vm ⊗ Vm given by the following coefficients . Rca db is a Yang-Baxter operator for each .m ≥ 1:

.

Rca db

⎧ 1, ⎪ ⎪ ⎪ ⎪ ⎨ y 2, ⎪ = ⎪ 1 − y 2, ⎪ ⎪ ⎪ ⎪ 0, ⎩

ifd = a ≥ b = c; if d = a < b = c; ifc = a < b = d; otherwise.

Directly, we see that the inverse of the operator is given by the coefficients:

(R−1 )ca db

.

⎧ 1, ⎪ ⎪ ⎪ ⎪ ⎨ y −2, ⎪ = ⎪ 1 − y −2, ⎪ ⎪ ⎪ ⎪ 0, ⎩

ifd = a ≤ b = c; if d = a > b = c; if c = a > b = d; otherwise.

The goal now is to obtain the second homology of . Rm . Recall that the chain modules .Cn and the boundary homomorphisms .∂n are given as in Definition 3.2.6, for which the graphical interpretation is given in Fig. 3.21. The main result is the following theorem which serves as the first nontrivial calculation of general (not set-theoretic) Yang-Baxter homology.

58

3 From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

Theorem 3.2.8 (PW) Let . R(m) be a unital Yang-Baxter operator giving the HOMFLYPT polynomial on level m as in Theorem 3.2.7. Then:

.

H2 (R(m) ) = k

1+( m 2)

) (m2 ) ( ) m−1 k k . ⊕ ⊕ 1 − y2 1 − y4 (

In particular, the ring .k can be either .Z[y ±1 ] or .Z[y]. Exercise 3.2.9 Compute .H3 (R(2) ). See [PW2] for conjecture about .H3 (R(m) ).

3.3 Exercises Exercise 3.3.1 Prove Lemma 3.1.2.

) = 3. Argue why the figure-eight knot cannot be Exercise 3.3.2 Show that tri( distinguished from the unknot by Fox 3-coloring. Use Fox-five colorings to conclude that the figure-eight knot is different from the trefoil knot. Exercise 3.3.3 1. Compute the fundamental group of Fox colorings of the square knot 31 #31 ; see Fig. 3.3. 2. Compute the fundamental group of Fox colorings Col(D) of the knot 52 . Exercise 3.3.4 Prove the following results about Fox n-colorability: 1. Show that Coln (D) forms a module over Zn . 2. Show that for p a prime number, col p (D) = pλ , for some number λ. Exercise 3.3.5 Prove Theorem 3.1.13. Exercise 3.3.6 Let (X, ∗) be a quandle. Then: 1. Verify the idempotency property of the operation ∗ holds, that is, a ∗ a = a. 2. Verify the right self-distributivity of the binary operation ∗, that is, (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c). 3. Verify the right distributivity of ∗ with respect to ∗, that is, (a ∗ b) ∗ c = (a ∗ c) ∗ (b ∗ c). Exercise 3.3.7 Show that any group G with conjugate operation a ∗ b = bab−1 is a quandle. Furthermore, show also that a ∗ b = b−1 ab.

Lecture 4 Goeritz and Seifert Matrices

In this lecture, we present two classical concepts introduced in the 1930s: the Goeritz matrix and the Seifert matrix. The first is constructed using the checkerboard coloring of a link diagram and the second using an oriented surface bounded by a link. We discuss several link invariants coming from the matrix including the determinant, the signature, the Alexander-Conway polynomial, and the Tristram-Levine signature.

4.1 Introduction Here we analyze two related matrices defined using link diagrams: the Goeritz and Seifert matrices, mostly following [Gor, GoLi, Prz21]. We start by showing that the Goeritz matrix can be used to define the signature and determinant of the link. We also analyze how .tk -moves change the matrix. These moves are important in Lecture 20, where they are used to define twisted torus links .T (k) (m, n), a special subset of which have considerable torsion in their Khovanov homology. Later in this lecture, we use the Seifert surface to construct the Seifert matrix and the Tristram-Levine signature which is a generalization of the classical signature.

4.2 History In 1930, Felix Frankl and Lev Pontrjagin first demonstrated that any knot bounds an orientable surface [FP]. Herbert Seifert gave a very simple construction of such a surface in 1934 [Sei]. The creation of the surface led to the Seifert matrix, as well as other invariants. In this lecture, we start with the work of Lebrecht Goeritz. In 1933, he showed how to associate a quadratic form with a diagram of a knot [Goe]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_4

59

60

4 Goeritz and Seifert Matrices

Although the signature of this quadratic form is not an invariant of the knot,1 the quadratic form can be used to establish other algebraic invariants. We then introduce the work of Seifert. Later, several applications of this surface, such as the TristramLevine signature, will be developed. Researchers have used a variety of approaches to study knots. In the first half of the twentieth century, combinatorial methods were preferred. For example, Reidemeister moves (see Fig. 3.1) were used to prove the existence of the Alexander polynomial, even though this was possible using the fundamental group. In the 1940s, Ralph Fox popularized techniques from algebraic topology using the fundamental group and coverings. Eventually, combinatorial methods were revived. The origins of their resurrection can be traced back to John H. Conway in [Con1] and were fully developed with Vaughan Jones’ breakthrough regarding Conway invariants and Louis Kauffman’s use of skein relations. Shortly, we will use a combinatorial approach to define the Goeritz matrix. In introducing the Seifert matrix, we use both combinatorial and topological methods.

4.3 Goeritz Matrix The first matrix producing link invariants was described in the letter of James W. Alexander to his mentor Oswald Veblen in 1919 [Ale2]; see Lecture 2. This matrix was generalized by Alexander in 1928 [Ale1] to construct, in particular, the polynomial now called the Alexander polynomial .Δ L (t). For .t = −1 we obtain the previous construction of Alexander and .Δ L (−1) gives an invariant called the determinant as it is the determinant of the Alexander matrix. In 1933, Goeritz constructed another matrix based on the link diagram also giving the determinant of the link as we describe in this section [Goe]. In fact, the Goeritz matrix is very closely related to the Alexander matrix for .t = −1, but it was understood only later.2 We turn now to the formal construction of the Goeritz matrix and outline a proof that the determinant of this matrix is an ambient isotopy invariant of links. This matrix was preceded by the Kirchoff matrix for electrical networks, established in 1847 [Kirc]. We recall now Tait’s construction of the plane graph from a link diagram (see Lecture 2). Definition 4.3.1 Let L be a connected diagram of a link. We begin by coloring the complement of the diagram in the projection plane .R2 , like a checkerboard, that is, color in black and white the regions into which the plane is divided by the diagram. Note that the black sections of the diagram give a surface, and the white sections give 1 In Definition 4.4.1, we show, after [GoLi], how to modify this signature so it becomes a link invariant. 2 Both matrices are related to the presentation of the first homology group of the double branched cover of .S 3 branched along the link. Fox coloring is also related to this homology [Prz13].

4.3 Goeritz Matrix

61

a different (complementary) surface. We assume that the unbounded region of .R2 \ L is colored white and it is denoted by . X0 , while the other white regions are denoted by . X1, . . . , Xn . Now, to any crossing p of L, we associate a number .η(p) = ±1 according to the convention described in Fig. 4.1.

= +1

= −1

Fig. 4.1: The convention for the value .η(p)

We use these .η values to construct the unreduced Goeritz matrix{ of the } n connected diagram, denoted by .G ' = G '(L). We define this matrix by .G ' := gi, j i, j=0 where:

.

gi, j

E ⎧ η(p) − ⎪ ⎪ Xi ⎪ ⎨ p connects ⎪ to X j E = ⎪ − gi,k ⎪ ⎪ ⎪ k=0,...,n; ⎩ k=i

for i = j, if i = j.

The matrix .G ' = G '(L) is called the unreduced Goeritz matrix of the diagram L. The reduced Goeritz matrix (or shortly Goeritz matrix) associated with the diagram L is the matrix .G = G(L) obtained by removing the first row and the first column of ' .G . To get a Tait’s signed graph .G(D) (see Sect. 2.4 of Lecture 2) from the diagram D with a checkerboard coloring, we have two choices. The first choice is to consider all white regions (w) and the second choice is to consider all black regions (b): (w) .G w (D) is obtained by choosing the white regions . X0, X1, . . . , Xn as vertices and the edges of .G w (D) are in bijection to crossings of D with the sign of the edge .e p being .η(p); see Fig. 4.1. (b) .G b (D) is constructed in a similar way but with vertices in bijection with the black regions of .R2 \ D. This agrees with the convention chosen by Tait. In fact, the plane graphs .G d (D) and .G b (D) are dual one to the other (see Definition 10.4.1).

62

4 Goeritz and Seifert Matrices

Two projections represent the same knot if one can be transformed into the other via a sequence of Reidemeister moves. The Goeritz matrices of the two diagrams are also related to each other via a relationship outlined in the following theorem. Theorem 4.3.2 ([Goe, KnP, Kyl]) Let us assume that . L1 and . L2 are two diagrams of a given link. Then the matrices .G(L1 ) and .G(L2 ) can be obtained one from the other in a finite number of the following operations on matrices: 1. .G ⇔ PGPT , where P is a matrix with integer entries and .det(P) = ±1. ] [ G 0 . 2. .G ⇔ 0 ±1 The first two conditions are sufficient for knots.3 ] [ G0 . 3. .G ⇔ 0 0 As a result, we know that . | det G| is an invariant of ambient isotopy of knots. The definition of the Goeritz matrix of a link was later generalized by Lorenzo Traldi [Tral]. The determinant can also be defined as the Alexander-Conway or Jones polynomial evaluated for .t = −1. See Chapter 9 of [Lic8]. We will prove this theorem using Reidemeister moves shortly. However, to gain familiarity and intuition with this link invariant, we first turn to a few examples. Example 4.3.3 Consider the right-handed trefoil knot diagram and the Hopf link diagram in Fig. 4.2. We calculate the Goeritz matrix and its determinant for the trefoil diagram (. D1 ) and the Hopf link (. D2 ). 1. For the right-handed trefoil knot diagram . D1 : [ ] [ ] 3 −3 ' .G (D1 ) = and G(D1 ) = 3 . −3 3 2. For the Hopf link diagram . D2 : [ ] [ ] 2 −2 ' .G (D2 ) and G(D2 ) = 2 . −2 2 Exercise 4.3.4 Suppose you exchange the roles of black and white in the checkerboard coloring described in the definition of the Goeritz matrix. Show that the determinant is unchanged.

3 More generally, it suffices for non-split links. A non-split link is by definition a link such that every graph of the link is connected.

4.3 Goeritz Matrix

63

Fig. 4.2: The right-handed trefoil knot and the Hopf link with checkerboard coloring

The checkerboard coloring was first used by Peter G. Tait [Tai3] in 1876. Following the convention of Cameron Gordon [Gor], we switched the role of black and white in our definition. We can say that Tait’s convention works well with a blackboard, while our convention works well with a white board. We now sketch a proof of Theorem 4.3.2 by examining how a Goeritz matrix changes under Reidemeister moves. Although the matrix does not depend on a fixed orientation, we assume that the diagram L is oriented and introduces an additional notation about crossings—see Fig. 4.3. Define: E . μ(L) := η(p). (4.1) p of type II

I

II

Fig. 4.3: Two types of oriented shaded crossings We recall that in Definition 4.3.1, we constructed, after Tait [Tai3], the graph G b (D) from the checkerboard coloring of the link diagram.4 See Fig. 4.4 for an example.

.

Let . β = B(L) denote the number of components of such a graph. From now on, let R be a Reidemeister move. We denote by .G1 the Goeritz matrix of L and by .G2 4 The construction is the inverse to the construction described in Lecture 2 where the link diagram was obtained from a signed plane graph. The graph we construct can be made into signed graph by using the convention of Fig. 4.1.

64

4 Goeritz and Seifert Matrices

the matrix of . R(L). Similarly, we set . μ1 = μ(L), . μ2 = μ(R(L)) and also . β1 = B(L), β2 = B(R(L)). If .G2 can be obtained from .G1 by a sequence of relations (1) and (2) from Theorem 4.3.2, then we write .G1 ∼ G2 .

.

Fig. 4.4: The trefoil diagram with its corresponding graph

4.4 Proof of Theorem 4.3.2 Proof 1. Let us consider the first Reidemeister move R1 . There are two cases depending on the coloring; see Fig. 4.5.

Case 1

+1

p Case 2 Fig. 4.5: The two cases for the Reidemeister type I move

(a) In Case 1, we note β1 = β2 , μ2 = μ1 + η(p). Notice that we have no new white regions and no new crossing exchanges between distinct white regions. Thus, the matrix is unchanged by this transformation. So we have G1 ∼ G2 .

4.4 Proof of Theorem 4.3.2

65

[ (b) In Case 2, we note β1 = β2 , μ1 = μ2 . We show G2 =

] G1 0 : 0 η(p)

The interactions between regions 0 − n are unchanged by this move. Thus, gˆi j = gi j for i < n, j < n. The entry for gnn now has an additional crossing of sign η(p). Therefore, gˆnn = gnn − η(p). Region Xn+1 does not interact with regions Xi where i < n. Thus, if i = n + 1, j < n, or i < n, j = n + 1, gˆi j = 0. Finally, region n + 1 has only one crossing of sign η(p), so gˆn+1,n+1 = −η(p). For gˆn,n+1 and gˆn+1,n , we have one crossing between the regions Xn+1 and Xn so these entries are η(p). To summarize: ⎡ g00 ⎢ ⎢ g10 ⎢ ⎢ . ' .G = ⎢ . 2 ⎢ . ⎢gn0 ⎢ ⎢ 0 ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ gn1 . . . gnn − η(p) 1 ⎥⎥ 0 ... 1 −η(p)⎥⎦ g01 g12 .. .

... ... .. .

g0n g1n .. .

0 0 .. .

The first row and column are eliminated to form G2 . The resulting matrix can be transformed via elementary row and column operations into the matrix: ] [ G1 0 . .G 2 = 0 −η(p) Using a result from linear algebra, | det(G2 )| = | det(G1 )|, as desired. 2. Next we consider the second Reidemeister move, R2 . Again there are two cases to consider, featured in Fig. 4.6. (a) Consider Case 1 in Fig. 4.6. Notice that no new white regions are formed by this move. Two new crossings are introduced into region Xn and Xn−1 . These crossings have opposite signs; thus, they are cancelled in the sum. Hence, the entry for gˆn,n−1 = gn,n−1 and μ1 = μ2 . Further, we can see that β1 β2 . Therefore, G2 ≈ G1 , and the determinant is unchanged by the transformation. (b) Consider Case 2 in Fig. 4.6. We have to consider two subcases. In each subcase, μ1 = μ2 , since the two new crossings are either both of type I or both of type II and always of opposite signs. (i) In the first case, we assume β1 = β2 . We start by forming G1' . By construction, we can eliminate the last row and column instead of the first without changing the determinant. So the matrix G can be written as:

66

4 Goeritz and Seifert Matrices

−1

−1

Case 1

Case 2

+1

+2

Fig. 4.6: The two cases for the Reidemeister type II move ⎡ g0,0 g0,1 g0,2 ⎢ ⎢ g1,0 g1,1 g1,2 ⎢ ' .G (K) = ⎢ . .. .. 1 ⎢ .. . . ⎢ ⎢gn−1,0 gn−1,1 gn−1,2 ⎣

. . . g0,n−1 ⎤⎥ . . . g1,n−1 ⎥⎥ ⎥. .. .. ⎥ . . ⎥ . . . gn−1,n−1 ⎥⎦

After the Reidemeister move is applied, the region Xn has been cut off from some number of exchanges, and these have been transferred to the new region Xn+2 . The region Xn+1 has been created. This new region has one positive and one negative crossing surrounding it. Thus, gˆn,n+1 = ±1, gˆn+1,n+2 = ∓1, and gˆn+1,n+1 = 0. In addition, the region Xn+1 has no interactions with other white regions in the knot. Hence, gˆn+1,i = gˆi,n+1 = 0 if i < n. Let -h = (h0,n, h1,n, . . . , hn,n, 1 fn,n ) denote the vector of exchanges between region Xn and all other regions after the second Reidemeister move is applied. Again, by construction we can cancel the row and column corresponding to region Xn+2 . So the matrix G2 can be written as: ⎡ g0,0 ⎢ ⎢ g1,0 ⎢ ⎢ .. ⎢ . .G 2 (R(K)) = ⎢ ⎢gn−1,0 ⎢ ⎢ h0,n ⎢ ⎢ 0 ⎣

g0,1 g1,1 .. . gn−1,1 h1,n 0

g0,2 . . . g1,2 . . . .. . . . . gn−1,2 . . . h2,n . . . 0 ...

g0,n−1 g1,n−1 .. .

h0,n h1,n .. .

gn−1,n−1 hn−1,n hn−1,n hn,n 0 1

0⎤⎥ 0⎥⎥ .. ⎥ . ⎥⎥ . 0⎥⎥ 1⎥⎥ 0⎥⎦

Using elementary row and column operations, we can therefore write ⎡G1 (K) 0 0⎤ ⎢ ⎥ ⎥ .G = ⎢ ⎢ 0 1 0⎥ . ⎢ 0 0 1⎥ ⎣ ⎦ ∗

4.4 Proof of Theorem 4.3.2

67

Thus, we have: .

| det(G∗ ) = | det(G1 )| = | det(G2 )|.

(ii) In the second case, we assume β2 = β1 − 1. In the argument for Case 1, we have assumed that the region xn+2 is distinct from xn . If this is not true, then the original diagram is not connected. To form the Goeritz matrix for a diagram that is not connected, you start by ordering the components. Then, the matrix is a block matrix with each block being the Goeritz matrix corresponding to a different component. In this case, the matrix G2 can be transformed via elementary operations into the matrix G1 . Thus, the determinant remains unchanged. 3. We now consider the third Reidemeister move, R3 (Fig. 4.7).

Fig. 4.7: Diagrams D and R3 (D); vertex p is the vertex between lower arcs of R3 (D) We see immediately that β1 = β2 . Next we should consider different orientations of arcs participating in R3 and two possibilities for the crossing p. However, we will always have μ2 = μ1 + η(p) and ] [ G1 0 . .G 2 ≈ 0 η(p) We leave it to the reader to check (see [Goe] and [Rei2]). See Fig. 4.8 for the diagrams used in two important cases. This concludes the proof of Theorem 4.3.2.

o

We now define the Trotter-Murasugi signature of a link using the Cameron Gordon and Richard Litherland approach [GoLi, Gor]. Recall that for a symmetric matrix, the signature of the matrix can be found by first diagonalizing the matrix and then counting the number of positive and negative diagonal entries. Thus, a size 4 matrix with signature (4, 1) has f our positive eigenvalues and one negative eigenvalue. We use the signature of the Goeritz matrix to define the signature of a link.

68

4 Goeritz and Seifert Matrices

Fig. 4.8: The third Reidemeister moves with two different shadings and signs for p

Definition 4.4.1 The Trotter-Murasugi signature for a link diagram L is defined by σ(L) = σ(G(L)) − μ(L) where σ(G(L)) is the signature of the Goeritz matrix of L and μ(L) is defined in Eq. (4.1). Corollary 4.4.2 The Trotter-Murasugi signature is independent of the link diagram, that is, σ(L) is an invariant of the link L, called the signature of the link. Corollary 4.4.3 Let us define nul(L) = nul(G(L)) + β(L) − 1, where β is the number of connected components of the link projection and nul(G(L)) is the nullity (i.e., the difference between the dimension and the rank of the matrix G(L)). Then nul(L) is an invariant of the link L and we call it the nullity (or defect) of the link. Corollaries 4.4.2 and 4.4.3 are proved by checking invariance of σ(L) and nul(L) under the Reidemeister moves. In 1985, Lorenzo Traldi introduced the generalized Goeritz matrix, which extends the definition of the determinant to links [Tral]. The signature and nullity of this matrix are invariants for links. Definition 4.4.4 Let L be a diagram of an oriented link. Then we define the generalized Goeritz matrix: ⎡ G O⎤ ⎢ ⎥ ⎥ . H(L) = ⎢ ⎢ A ⎥, ⎢O B ⎥ ⎣ ⎦ where G is a Goeritz matrix of L and the matrices A and B are defined as follows. The matrix A is diagonal of dimension equal to the number of type II crossings and

4.4 Proof of Theorem 4.3.2

69

the diagonal entries equal to η(p), where p’s are crossings of type II. The matrix B is of dimension β(L) − 1 with all entries equal to 0. The lemma below follows immediately from the proof of Theorem 4.3.2. Lemma 4.4.5 ([Tral]) If L1 and L2 are diagrams of two isotopic oriented links, then H(L1 ) can be obtained from H(L2 ) by a sequence of the following elementary equivalence operations: 1. H ⇔ PHPT , where P is a matrix with integer entries and det(P) = ±1. ⎡H O ⎤ ⎢ ⎥ 2. H ⇔ ⎢⎢ 1 ⎥⎥ . ⎢O −1⎥ ⎣ ⎦ From this lemma, we obtain the following corollary, which defines the determinant of a link. √ Corollary 4.4.6 The determinant det(iH(L)), where i = −1, is an invariant of a link L, called the determinant of the link, Det L . Moreover, σ(H(L)) = σ(L) and nul(H(L)) = nul(L). So far, we have defined and proven three new link invariants—the determinant, signature, and nullity of a link. In Exercise 4.4.9, we show that if Det L = 0, then it determines the signature of L modulo 4. We start from the simple example of the torus link T(2, k). Example 4.4.7 Consider a torus link of type (2, k), T2,k . It is a knot for odd k and a link of two components for k even; see Fig. 4.9. [ ] k −k , and thus the Goeritz matrix of The matrix G ' of T2,k is then equal to −k k the link is G = [k]. Moreover, β = 1 and μ = k because all crossings are of type II. Therefore, for k = 0, σ(T2,k ) = σ(G) − μ = 1 − k and nul(T2,k ) = nul(G) = 0. The generalized Goeritz matrix H of the knot T2,k is of dimension k + 1 and it is equal to: ⎡k O ⎤⎥ ⎢ ⎢ −1 ⎥ ⎢ ⎥ ⎢ ⎥ −1 .H = ⎢ ⎥. ⎢ ⎥ .. ⎢ ⎥ . ⎢ ⎥ ⎢O ⎥ −1 ⎣ ⎦ Therefore, Det L = det(iH) = (−1)k i k+1 k = i 1−k k. Notice also that i σ(T2, k ) = DetT2, k |DetT | ; compare Exercise 4.4.9. 2, k

Let us note that if we connect black regions of the plane divided by the diagram of the link by half-twisted bands (as indicated in Fig. 4.10), then we get a surface in R3 (and in S 3 ), the boundary of which is the given link; we denote this surface by Fb

70

4 Goeritz and Seifert Matrices

T 2,k

X0

X1

k halftwists

Fig. 4.9: The torus link, T(2, k)

and call the Tait surface of a link diagram. 5 If for some checkerboard coloring of the plane the constructed surface has an orientation, which yields the given orientation of the link, then this oriented diagram is called a special diagram.

Fig. 4.10: Creating a surface locally

Exercise 4.4.8 Prove that an oriented diagram of a link is special if and only if all crossings are of type I for some checkerboard coloring of the plane. Conclude from this that for a special diagram D, we have σ(D) = σ(G(D)). Exercise 4.4.9 Show that any oriented link has a special diagram. Conclude from this that for any oriented link L, one has Det L = i σ(L) |Det L |; compare Example 4.9.9. We now turn our attention to n-moves. Definition 4.4.10 An n-move is a local change of an unoriented link diagram by n-twists as seen in Fig. 4.11. For oriented links, a tn -move is a modification of an oriented link by n-twists as seen in Fig. 4.12.

5 This definition can be generalized to any Kauffman state of the diagram; see Lectures 5 and 19; compare Definition 2.1 of [AP].

4.4 Proof of Theorem 4.3.2

71

n− move

D0

Dn

n right handed half twists

Fig. 4.11: n-moves on a link diagram (D0 → Dn ) −move

−−−−−−→

¯2 −move

···

···

−−−−−−−→

-half twists

2 -half twists

Fig. 4.12: Oriented tn -move and t¯2k -move after [Prz2] These moves are important in later lectures where they are used to define T (k) (m, n) twisted torus links. Figure 4.13 shows the result of applying an n-move to a checkerboard coloring.

.. .

n-move

∞

Fig. 4.13: Ln is obtained from L = L0 by an n-move, and L∞ We can relate the Goeritz matrices of Ln and L∞ using the lemma below. We assume the black regions are chosen as in Fig. 4.11. The white region in R2 \ L∞ is divided into two regions in R2 \ L. We have the following relationship between the Goeritz matrix of Ln and that of L∞ . Lemma 4.4.11

] G(L∞ ) α , .G(Ln ) = αT q + n [

where α is a vector with dimension equal to the number of white regions in G(L∞ ) and q is any integer value. Corollary 4.4.12 1. det(G(Ln )) − det(G(L0 )) = n · det(G(L∞ )). 2. σ(G(L0 )) ≤ σ(G(Ln )) ≤ σ(G(L0 )) + 2 for n ≥ 0. 3. |σ(G(Ln )) − σ(G(L∞ ))| ≤ 1. Further, σ(G(Ln )) = σ(G(L∞ ) if and only if rank(G(Ln )) = rank(G(L∞ )) + 2.

72

4 Goeritz and Seifert Matrices

By orienting the link L = L0 , we can use this corollary to obtain useful properties of σ(L) and σ(Ln ). Corollary 4.4.13 ([Prz5]) 1. Assume that L0 is oriented such that its strings are parallel and Ln be obtained from L0 by a tn -move. (

) Then: .

n − 2 ≤ σ(L0 ) − σ(Ln ) ≤ n.

2. Assume that L0 is oriented such that its strings are antiparallel and that ¯ move. n = 2k is an even number. Let L2k be obtained from L0 by a t2k (

) Then: 0 ≤ σ(L2k ) − σ(L0 ) ≤ 2.

.

3. In particular, for n = 2 we get the classical result of Murasugi [Mura1], Conway ([Con1] page 340), and C. A. Giller (Remark 3 of [Gi]), that is, for any knot, K: 0 ≤ σ(K− ) − σ(K+ ) ≤ 2.

.

4.5 Signature of Alternating Links and Quasi-alternating Links In this section, we describe the combinatorial formula found by Traczyk [Tra5] following a paper by the first author [Prz21]. Traczyk’s formula allows to find the signature of an alternating diagram as long as we know the writhe of the diagram .w(D), and the number . d+ − d− . Here . d+ is the number of .+ crossings in an arbitrary fixed checkerboard coloring, and .d− is the Enumber of .− crossings (see convention described in Fig. 4.1). That is, .d+ − d− = η(p). p

Theorem 4.5.1 ([Tra5]) If D is a reduced6 alternating diagram of an oriented link, then: .σ(D) = −w(D) + d+ − d− . We refer to [Tra5] or [Prz21] for a detailed proof. In the next few exercises, we ask the reader to show facts leading to the theorem, so that with some effort, the reader should be able to reconstruct a proof. Exercise 4.5.2 Show that if a diagram has only nugatory crossings, then it represents a trivial link. 6

Reduced means that no crossing D is nugatory. See Lecture 2.

4.5 Signature of Alternating Links and Quasi-alternating Links

73

Exercise 4.5.3 If D is a reduced alternating diagram of an oriented link, then for any crossing p of D, we have: p

p

(1) .σ(D) = σ(D0 ) − sgn(p), where . D0 denotes the diagram obtained from D by smoothing the crossing p along the orientation of D. p

p

p

(2) .σ(D) = σ(D∞ ) − 21 sgn(p)(w(L0 ) − w(L∞ )), where . L∞ is a diagram obtained from D by smoothing the crossing p against its orientation and choosing for the resulting diagram arbitrary but fixed orientation. One of the oldest results in knot theory is the following observation by Listing [Lis]: Exercise 4.5.4 Let D be an unoriented link diagram. Let us mark the corners of each crossing of D by .λ (laeotropic) and .δ (dexiotropic) as shown in Fig. 4.14.7 Show that if D is a connected diagram, then it is an alternating diagram if and only if the regions are monochromatic. That is, .R2 \ D has only regions with all markers .λ or all corners of a region have markers .δ. In particular, for alternating diagram, we have 2 .λ − δ checkerboard coloring of the regions of .R \ D.

B A

A B

Fig. 4.14: Marking corners of a crossing The conditions of Exercise 4.5.3 are very close to the condition used more recently by P. Ozsvath and Z. Szabo to introduce the notion of quasi-alternating links [OS]; the concept has nice consequences in the theory of Khovanov homology (see Lectures 19 and 20). Definition 4.5.5 ([OS]) The family of quasi-alternating links is the smallest family of links which satisfies the following conditions: (i) The trivial knot is quasi-alternating. (ii) If L is a link which admits a crossing such that (1) both smoothings (. L0 and . L∞ ) are quasi-alternating, and (2) . |Det L | = |Det L0 | + |Det L∞ |, then L is quasi-alternating. 7

Kauffman marks the corner by A and B as in Fig. 4.14, and it is an important ingredient of the Kauffman bracket construction.

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4 Goeritz and Seifert Matrices

The crossing used in the definition is called a quasi-alternating crossing of L. We formulated the condition for a link to be quasi-alternating in the language of the absolute value of the determinant. It has the advantage of working only with unoriented diagrams. However, in an equivalent way, we can work with the signature condition. Exercise 4.5.6 The following two conditions are equivalent, provided that the determinants of . L0 and . L∞ are not equal to zero (we work with an arbitrary but fixed crossing p). (1) . |Det L | = |Det L0 | + |Det L∞ |. (2) .σ(D) = σ(D0 ) − sgn(p), and .σ(D) = σ(D∞ ) − 12 sgn(p)(w(L0 ) − w(L∞ )). We now turn our attention to Seifert surfaces. These surfaces are not link invariants themselves, but the Seifert matrix is derived from these surfaces and gives rise to new invariants such as the Tristram-Levine signature.

4.6 Seifert Surfaces In this section, we define the Seifert surface and the genus of a link. Although the Seifert surface is not a link invariant, the genus is. We culminate the section with a definition of the connected sum of links and results pertaining to the genus of this construction. Historically, Frankl and Pontrjagin first demonstrated in 1930 that any knot bounds an orientable surface [FP]. Later, H. Seifert found a simple construction of this surface in [Sei] and developed several applications of the surface which we discuss in this section. Definition 4.6.1 A Seifert surface8 of an unoriented link . L ⊂ S 3 is a compact, connected, orientable 2-manifold such that .∂S = L. If a link is oriented, then its Seifert surface S is assumed to be oriented so that the orientation agrees with that of L. The genus of a link is the minimal genus of a Seifert surface of L. Notice that for knots the genus does not depend on the orientation. However, for links the unoriented link can have a smaller genus than the same link with orientation—see Remark 4.6.5. The surface itself is not a knot invariant. Depending on the projection chosen, several surfaces are possible for a given knot. However, the genus is an invariant of knots and links. 8

Kauffman in [Kau4, Kau8] uses the term Seifert surface to describe the surface obtained from an oriented link diagram by the Seifert algorithm (Algorithm 4.6.4) and the term spanning surface for an oriented surface bounding a link (our Seifert surface of Definition 4.6.1). In [BE] the name Frankl surface is used for any oriented or unoriented spanning surface.

4.6 Seifert Surfaces

75

Example 4.6.2 A Seifert surface of a trefoil knot is pictured in Fig. 4.15.

Fig. 4.15: The trefoil diagram with its corresponding Seifert surface. Note that the genus of this surface is 1 We have the following theorem which shows the genus is well-defined. Theorem 4.6.3 ([FP, Sei]) Every link in .S 3 bounds a Seifert surface. If the link is oriented, then there is a Seifert surface with an orientation determined by the orientation of the link. The proof of Theorem 4.6.3 is given the following algorithm of Seifert’s. Algorithm (Seifert) [Sei] Consider a fixed diagram D of an oriented link L in .S 3 . In the diagram, there are two types of crossings; in a neighborhood of each of the crossings, we make a modification of the link (a smoothing) according9 to Fig. 4.16.

−→

←−

Fig. 4.16: Examples of smoothing a crossing according to its orientation After smoothing all crossings of D, we obtain a family of disjoint oriented simple closed curves in the plane, called Seifert circles by R. Fox [Fox], and denoted by . D - . Each of the curves of . D - bounds a disk in the plane; the disks do not have to S S be disjoint (they can be nested). Now we make the disks disjoint by pushing them slightly up above the projection plane. We start with the innermost disks (i.e., disks 9

This modification is the special case of the Kauffman bracket state modification discussed in Lecture 19 in connection with Khovanov homology.

76

4 Goeritz and Seifert Matrices

without any other disks inside) and proceed outward (i.e., if . D ' ⊂ D, then . D ' is pushed above D; see Fig. 4.17).

Fig. 4.17: Disks used to construct Seifert surfaces

The disks are two-sided so we can assign the signs .+ and .− to each of the sides of a disk according to the following convention: the sign of the “upper” side of the disk is .+ (respectively .−) if its boundary is oriented counterclockwise (respectively clockwise). See Fig. 4.18.

Fig. 4.18: Signs of an oriented disk Now we connect the disks together at the original crossings of the diagram D by half-twisted bands so that the 2-manifold which we obtain has L as its boundary. Since the “.+ side” is connected to another “.+ side,” it follows that the resulting surface is orientable. Moreover, this surface is connected if the projection of the link is connected (e.g., if L is a knot). If the surface is not connected, then we join its components by tubes (see Fig. 4.19) in such a way that the orientation of components is preserved.

Fig. 4.19: Connecting two disjoint components of a surface by a tube

4.6 Seifert Surfaces

77

Remark 4.6.5 If the link L has more than one component, then the Seifert surface, which we constructed above, depends on the orientation of components of L. This can be seen in the example of a torus link of type .(2, 4); see Fig. 4.20.

+ +

−

+

+

−

+ (a)

(b)

Fig. 4.20: Different orientations result in different Seifert Surfaces

The Seifert surface from Fig. 4.20a has genus 0, while the surface from Fig. 4.20b has genus 1. Therefore, the link L has genus 0 (as an unoriented link). We now turn our attention to the genus of a link, noting that there is a formula for it in terms of crossings, components, and Seifert circles. Proposition 4.6.6 If a projection of a link L is connected (e.g., if L is a knot), then the surface, from Algorithm 4.6.4, is unknotted, that is, the complement of the surface in .S 3 is a handlebody. The genus of the handlebody is equal to .c + 1 − s, and the Euler characteristic is equal to .s − c, where c denotes the number of crossings of the projection and s the number of Seifert circles. Proof The complement of the plane projection of L is a 3-ball (some refer to this as a 3-disk) with .c + 1 handles (the projection of L cuts the projection plane (or 2-sphere) into .c + 2 regions). Furthermore, by adding s 2-disks in the construction of the Seifert surface, we cut s of the handles; thus, the result remains a 3-ball with .c + 1 − s handles. See Fig. 4.21 for a demonstration with the trefoil knot. The Euler characteristic of the obtained handlebody is equal to .1 − (c + 1 − s) = s − c. o Exercise 4.6.7 Show that the complement of the plane graph in .S 3 is a handlebody. Exercise 4.6.8 A knot K in .S 3 is trivial if and only if its genus is equal to 0. Exercise 4.6.9 Let L be a link with n components and . D L its diagram. Let c denote the number of crossings in . D L and let s be the number of Seifert circles. Prove that the genus of the resulting Seifert surface, .g(S), is equal to:

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4 Goeritz and Seifert Matrices

Fig. 4.21: The complement of the plane projection of the trefoil knot is shown on the left. During the construction of the Seifert surface of the trefoil, two 2-handles are cut, as shown on the right

.

g(S) = p −

s+n−c , 2

where p is the number of connected components of the projection of . D L . Check that the Euler characteristic of S, for . p = 1, is equal to .s − c so it agrees with the Euler characteristic of the handlebody described in Corollary 4.6.6. The next exercise describes a natural generalization of Theorem 4.6.3 to an arbitrary 3-dimensional manifold. Exercise 4.6.10 Let M be a 3-dimensional manifold and L an oriented link in M which is homologically trivial. Then there is an oriented surface F embedded in M such that .∂F = L. Hint: Construct a map . f : M → S 1 which is an epimorphism on the fundamental group. Modify f so that . f −1 (b) where b is a fixed point in .S 1 , which is an oriented surface (see [Stal, Hem, Jac, JaPr]). The conclusion of the exercise is a starting step in finding the q-homology skein module of a 3-manifold (see Lecture 11). We next investigate the connected sum of knots, with the goal of establishing a formula for its genus. These facts culminate in a discussion relating these properties to the trefoil knot. Recall that the regular neighborhood is the equivalent of the tubular neighborhood except that the former is for the piecewise linear category and the latter is for the smooth category. Definition 4.6.11 ([Schu1, BZ]) Assume that .K1 and .K2 are oriented knots in .S 3 . A connected sum of knots (or composition of knots), .K = K1 #K2 , is a knot K in .S 3 which is constructed as follows: First, for .i = 1, 2 choose a point . xi ∈ Ki and its regular neighborhood .Ci in the pair 3 3 3 .(S , Ki ). Then, consider a pair .((S \int(C1 )∪φ S \int(C2 ), (K1 \int(C1 )∪φ K2 \int(C2 )),

4.6 Seifert Surfaces

79

where .φ is an orientation reversing homeomorphism .∂C1 → ∂C2 which maps the end of .K1 ∩ (S 3 \ int(C1 )) to the beginning of .K2 ∩ (S 3 \ int(C2 )) (and vice versa). (Notice that notions of beginning and end are well-defined because .K1 and .K2 are oriented.) We see that .(S 3 \ int(C1 ) ∪φ (S 3 \ int(C2 ) is a 3-dimensional sphere and .K1 #K2 = (K1 \int(C1 )∪φ (K2 \int(C2 )) is an oriented knot, called the connected sum of .K1 and .K2 . Although this construction depends on many choices, the connected sum of knots is well-defined. The connected sum of oriented links follows a similar definition, but depends on the choice of components on which the sum is performed. We now have the appropriate framework to consider what it means for a knot to be prime. Definition 4.6.12 A prime knot is one that is not a connected sum of nontrivial knots. That is, if K is a prime knot and .K = K1 #K2 , then either .K1 or .K2 is the trivial knot. The set of knots under the connected sum operation forms a nice algebraic structure. Theorem 4.6.13 ([Schu1]) Any knot in .S 3 admits a decomposition into a finite connected sum of prime knots. Further, this decomposition is unique up to the order of factors, making knots a unique factorization commutative monoid under the connected sum operation. The weak version of the unique prime factorization of links with respect to connected sum was proven in 1958 by Youko Hashizume [Hash]. The specification that .φ is an orientation reversing homeomorphism is significant. If we take .K1 as the right-handed trefoil knot .31 , and .K2 as the mirror image of the trefoil knot .31 , then the connected sum .31 #31 is the square knot, whereas the connected sum of .K1 with itself .31 #31 is the granny knot. These can be distinguished using the Jones or HOMFLYPT polynomials. Remarkably, the genus of knots is additive over the connected sum, as discussed in the following theorem. Theorem 4.6.14 ([Schu2, Schu3]) The genus of knots in .S 3 is additive, that is: .

g(K1 #K2 ) = g(K1 ) + g(K2 ).

Proof A proof of this theorem can be found in Chapter 4 of C. Adams book [Ada] o and in Chapter 2 of W. B. R. Lickorish [Lic8]. The discussion of the genus and the connected sum allows us to conclude the following: Corollary 4.6.15 The trefoil knot is nontrivial and prime.

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4 Goeritz and Seifert Matrices

Proof The trefoil is nontrivial because the determinant of its Goeritz matrix is 3 and the determinant of the trivial knot’s Goeritz matrix is 1. Thus, the determinant distinguishes the trefoil from the unknot. By Corollary 4.6.8 the genus of the trefoil must be positive. Thus, Fig. 4.15 establishes that the genus of the trefoil is 1. Suppose the trefoil could be written as the connected sum of two nontrivial knots .K1 and .K2 . Then .g(31 ) = g(K1 #K2 ) = g(K1 ) + g(K2 ). Since both .K1 and .K2 are nontrivial, by Corollary 4.6.8, .g(K1 ) = 0 and .g(K2 ) = 0. Thus, we would have .g(31 ) > 1, which is a contradiction. o In summary, we have given a definition and algorithmic construction for the Seifert surface given an oriented link diagram. A new link invariant was established: the minimal genus over all such surfaces. We established several results pertaining to a formula for the genus, and we applied these results to the trefoil, establishing that it is nontrivial and prime. In the next section, we discuss the Seifert matrix of a knot. This matrix can be used to find the determinant of a link and its generalization to the Alexander polynomial. The matrix is also used to define the classical signature and its generalization to the Tristram-Levine signature.

4.7 The Seifert Form and the Seifert Matrix We now turn our attention to the Seifert matrix of a knot in which its definition relies heavily on the notion of linking number and Seifert form. We start by introducing the linking number .lk(J, K) for any pair of disjoint oriented knots J and K. Recall the usual diagrammatic definition of the linking number: For two knots .K1 and .K2 , the sum of all positive crossings between .K1 and .K2, minus the sum of all negative crossings between the two knots is equal to twice the linking number. We provide a topological definition and show that this definition agrees with the diagrammatic definition. Definition 4.7.1 The linking number .lk(J, K) is an integer such that .[J] = lk(J, K)[m], where .[J] and .[m] are homology classes in .H1 (S 3 \ K) of the oriented curve J and the meridian m of the oriented knot K, respectively. Chapter 5 part D of [Rol] gives many other equivalent definitions of the linking number—including the original one given by Gauss in 1833 (see Lecture 2). The following lemmas establish that this definition agrees with the commonly considered diagrammatic definition. Lemma 4.7.2 Let .S ⊂ S 3 \ int(Vk ) be a Seifert surface of a knot K (more precisely, its restriction to .S 3 \ int(Vk )), such that its orientation determines the orientation of . ∂S compatible with that of the longitude l. Then .lk(J, K) is equal to the algebraic intersection number of J and S.

4.7 The Seifert Form and the Seifert Matrix

81

Proof We begin by formally describing the orientation of the boundary of an oriented manifold M. Let . x ∈ M. We have a basis of the tangent space10 .Tx ∂ M, . {v2, v3, . . . , vn }, and a normal vector .n ¯ for .∂ M in M which is directed perpendicular to the boundary. The collection of vectors .{v2, v3, . . . , vn, } defines the orientation of ¯ v2, . . . , vn } defines the orientation of .Tx M. the .Tx ∂ M if .{ n, Let m denote the meridian of the knot. Using our definition of the orientation of the Seifert surface S, m intersects S in exactly one point with an algebraic intersection number of .+1. Suppose the algebraic intersection number of the curve J and S is o equal to i. Then .[J] = i[m]. Thus, .i = lk(J, K). Lemma 4.7.3 Let us consider a diagram of a link . J ∪ K consisting of two disjoint oriented knots J and K. We assume that the orientation of .S 3 = R3 ∪ ∞ is induced by the orientation of the plane containing the diagram of . J ∪ K and the third axis which is directed upward. Now, to any crossing of the diagram where J passes under K, we assign .+1 in the case of a positive crossing and .−1 in the case of a negative crossing. Then, the sum of all numbers assigned to such crossings is equal to twice the linking number .lk(J, K), and thus the topological definition of the linking number given above is consistent with the diagrammatic definition. Proof Consider a Seifert surface constructed from the diagram D of the knot K. Assume the knot J is placed above this surface, except in small neighborhoods where J passes under K. The sign of the intersection of J with this surface coincides with the sign associated with the crossing by the convention described in the lemma. o The linking number may be defined for any two disjoint 1-cycles (e.g., oriented links) in .S 3 . For example, as a definition, we may use the condition from Lemma 4.7.2. That is, if .α and . β are disjoint 1-cycles (e.g., oriented links) in .S 3 , then .lk(α, β) is defined as the intersection number of .α with a 2-chain in .S 3 (e.g., Seifert surface) whose boundary is equal to . β. Exercise 4.7.4 Prove that .lk(α, β) is well-defined, i.e., it does not depend on the 2-chain whose boundary is . β. Exercise 4.7.5 Show that .lk is symmetric and bilinear, i.e., .lk(α, β) = lk(β, α), lk(α, nβ) = n · lk(α, β), and if a cycle . β ' is disjoint from .α, then .lk(α, β + β ') = lk(α, β) + lk(α, β ').

.

Exercise 4.7.6 Prove that if . β and . β ' are homologous in the complement of .α, then ' .lk(α, β) = lk(α, β ). The definition of the Seifert matrix relies on that of the Seifert form. Let .S × [−1, 1] be a regular neighborhood of S in .S 3 . For a 1-cycle x in .int(S), we can consider a cycle . x + (respectively . x − ) in .S × 1 (respectively .S × −1) which is obtained by shifting 10

For more on tangent spaces, see Appendix B.

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4 Goeritz and Seifert Matrices

the cycle x to .S × 1 (respectively .S × −1).11 We note that the sides of S are uniquely defined by the orientations of the link L and .S 3 . Definition 4.7.7 The Seifert form of the oriented link L is a bilinear form defined on the first homology of the Seifert surface . f : H1 (int(S)) × H1 (int(S)) → Z such that + . f (x, y) = lk(x , y). Lemma 4.7.8 The Seifert form f is a well-defined bilinear form on the .Z-module (i.e., abelian group) .H1 (S). Proof The result follows from Exercises 4.7.5 and 4.7.6.

o

The homology of a compact surface of genus g and .d boundary components, denoted by .Fg,d , is the free group of .2g+d−1 generators, that is, .H1 (Fg,d ) = Z2g+d−1 . Definition 4.7.9 Let .e1, e2, . . . , e2g+com(L)−1 be a basis of .H1 (S). Then the Seifert matrix .V = (vi j ) is defined on this basis by: .

vi, j = lk(ei+, e j ).

Then for . x, y ∈ H1 (S), we have . f (x, y) = xT V y. We use the convention that coefficients of a vector are written as a column matrix. A different basis will give a matrix that is similar to the one defined on the basis above. That is, a change of the basis in .H1 (S) results in the change of the matrix V to a similar matrix .PT V P, where .det P = ±1.

4.7.1 Examples of Computing Seifert Matrices from Seifert Surfaces In practice, when computing the Seifert matrix of a Seifert surface S from a given oriented link diagram L, we choose our basis to be represented by simple closed curves + .α1, α2, . . . , αg , .. . . , α2g+com(L)−1 in S. Then we compute the i jth entry .lk(α , α j ), usi ing the diagrammatic definition of the linking number with the help of the pictorial descriptions shown in Figs. 4.22 and 4.23.

11

We call the cycle . x + the push-off of x.

4.7 The Seifert Form and the Seifert Matrix

−

83

−

+

+

passing (a) local depiction of a diagram of passing (b) local depiction of a diagram of by a twist in the surface. under by a twist in the surface. over

Fig. 4.22: A crossing between .αi and .α j resulting from a twist in the surface. This behavior persists even when the push-offs—.αi+ and .α+j —are considered in the computation of the linking number

+

+

+

+ +

+

−

−

+

+

+ +

(a) A local depiction of a Seifert surface with "+" side and + in × {1}.

−

− −

−

− −

(b) A local depiction of a Seifert surface with "-" and + in × {1}.

Fig. 4.23: To compute .lk(αi+, α j ), we consider the curves in .S × [−1, 1] such that .αi+ is embedded in .S × {1} in order to determine the crossing of the intersection point

Before we continue with concrete examples, we encourage readers who are unfamiliar with this topic to complete the following exercise. Exercise 4.7.10 Create a model of the Seifert surface of the Hopf link by cutting out two disks of clear paper. Connect these two disks by two half-twisted strips, and then draw the simple closed curve that represents the generator of .H1 (S). The curve is actually embedded into the surface—which is why clear paper is required for proper visualization. Include orientations. To see .α+ , lie a piece of string along .α and then tie this string into a loop—see Fig. 4.24. Create a similar model for the trefoil knot.

Example 4.7.11 Let L be the Hopf link whose Seifert surface is shown in Fig. 4.24. For the Hopf link, .com(L) = 2, and .g(L) = 0. Therefore, we have .2g+com(L)−1 = 1 element, e, in the basis of .H1 (S). Let .α be a simple closed curve such that .[α] = e. We compute .lk(α+, α). We start with the picture of the Hopf link’s Seifert surface, S, with .α and a slightly displaced .α, denoted by .α˜ as shown in Fig. 4.24. When viewed in .S 3 , two crossings take place within the twists—see Fig. 4.23. Working in

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4 Goeritz and Seifert Matrices

R3 ∪ {∞}, we delete the surface S. We then project .α+ ∪ α onto the plane—.R2 —and consider them as link diagrams; see Fig. 4.24. Let .{ci } denote the set of all crossings E between .α+ and .α. We see in Fig. 4.24 that .lk(α+, α) = 12 sgn(ci ) = 1. Since S is

.

ci

an [ ]annulus, .H1 (S) has only one generator, and its Seifert matrix is the .1 × 1 matrix . 1 .

+

−

˜

(b) The Hopf link with and slightly displaced, denoted by ˜ .

(a) The Hopf link with . +

(c) View

+

and

as a link diagram.

Fig. 4.24: Examples of the basis curve .α in the Seifert surface of the Hopf link

Remark 4.7.12 In general, forEany .α a simple closed curve on the Seifert surface, we can compute .lk(α+, α) = − 12 sgn(pi ) where the sum is taken over all crossings of the diagram traversed by .α.

pi

Example 4.7.13 Consider the torus link .T (2, n) with antiparallel orientation as shown in Fig. 4.25. Notice that this is a generalization of Example 4.7.11 since .T (2, 1) is the Hopf link. The Seifert surface S of the torus link .T (2, n) is an annulus, as seen in Fig. 4.25. Therefore, .H1 (S) has only one generator, and by using Remark 4.7.12, we see that the Seifert matrix is the .1 × 1 matrix .[n].

2 -half twists (a) The torus link

2 -half twists (2

).

(b) A Seifert surface for the torus link

(2

).

Fig. 4.25: The Seifert surface of the torus link of type .(2, n), denoted by .T (2, n) with orientation shown in (a) is an annulus with 2n half-twists as shown in (b)

4.7 The Seifert Form and the Seifert Matrix

85

+

+

(a)

+

+

(b)

(c)

Fig. 4.26: Computing the off-diagonal entries of the left-handed trefoil’s Seifert matrix. The black dot in (a) represents the intersection point in the surface. In (b), we see the surface together with the push-offs .α+ (top) and . β+ (bottom). In (c), we view the curves and their push-offs as link diagrams

Example 4.7.14 Let L be the right-handed trefoil knot shown in Fig. 4.20b, then com(L) = 1, and as we saw in Remark 4.6.5, .g = 1. Therefore, we have .2g + com(L) − 1 = 2 elements in the basis of .H1 (S)—in fact S is a torus with a hole. Let .α1 and .α2 be two simple closed curves such that .[α1 ] = e1 and .[α2 ] = e2 . We will compute .lk(α1+, α2 ). We start with the diagrammatic picture of L with .α1 and 2 .α2 as shown in Fig. 4.26a. Notice that as link diagrams in .R , .α2 crosses over .α1 via the twist in the surface, see Fig. 4.26c. In the figure, the black dot represents an intersection point where the crossing information is unknown. We use Fig. 4.23 to draw L with .α1+ and .α2 . We see in Fig. 4.26c that .lk(α1+, α2 ) = 0. We compute + + .(α , α1 ) in a similar way and .lk(a , ai ) using the formula from Remark 4.7.12. After i 2 [ ] −1 0 computing the last three linking numbers, we obtain the Seifert matrix of L, . . 1 −1 .

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4 Goeritz and Seifert Matrices

+ +

+ −

Fig. 4.27: The left side shows the usual presentation of the Seifert surface of the figure-eight knot. The right side features a view of the deformed surface to illustrate the genus and curve intersections

Example 4.7.15 Let L be the figure-eight knot with Seifert surface S, as shown in Fig. 4.27. Since .com(L) = 1 and S has genus .g = 1, then we have two elements in the basis of .H1 (S). Let .α1 and .α2 be two simple closed curves such that .[α1 ] = e1 and .[α2 ] = e2 , as shown in Fig. 4.27. Then[ the Seifert matrix of a Seifert surface S ] 1 −1 computed in the basis .{e1, e2 } is equal to . . 0 −1 Example 4.7.16 Consider the pretzel link of type .(n1, n2, n3 ) denoted by .Pn1,n2,n3 . Then let L be the pretzel knot of type .(1, 3, 5), denoted by .P1,3,5 , with Seifert surface S, as shown in Fig. 4.28. Since .com(L) = 1 and S has genus .g = 1, then we have two elements in the basis of .H1 (S)—in fact S is actually a torus with a hole. Let .α1 and .α2 be two simple closed curves such that .[α1 ] = e1 and .[α2 ] = e2 , as shown in Fig. 4.28. Then the [ Seifert ] matrix of a Seifert surface S computed in the basis −2 1 . . {e1, e2 } is equal to . 2 −4 With some practice, one should be able to find the Seifert form efficiently, and we encourage the reader to compute more examples and develop some rules; for example, Remark 4.7.12 and the following generalization for pretzel knots of type .(2k 1 + 1, 2k 2 + 1, 2k 3 + 1). Exercise 4.7.17 Show that the pretzel link .P1,1,1 is the right-handed trefoil knot, and the link .P2,1,1 is the figure-eight knot. Exercise 4.7.18 Show that the Seifert matrix of a Seifert surface S of the pretzel knot .P2k1 +1,2k2 +1,2k3 +1 , computed in the basis 4.28 ] [ analogous to that of Example −k1 − k2 − 1 k2 . . {[α1 ] = e1, [α2 ] = e2 } of . H1 (S), is equal to . k2 + 1 −k2 − k3 − 1

4.7 The Seifert Form and the Seifert Matrix

87

Fig. 4.28: .P1,3,5 the pretzel knot of type .(1, 3, 5)

4.7.2 S-equivalence It should be clear that the Seifert matrix completely depends on the diagram chosen for the knot or link. The Seifert matrix is therefore not an invariant; however, relationships can be established between the various Seifert matrices for a given knot.

Definition 4.7.19 We call two matrices S-equivalent if one can be obtained from the other by a finite number of the following modifications: • . A ⇔ P APT where .det P = ±1. ⎡ A α 0⎤ ⎡ A 0 0⎤ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ • . A ⇔ ⎢ 0 0 1⎥ and . A ⇔ ⎢⎢ β 0 0⎥⎥ where .α is a column and . β is a row. ⎢ 0 0 0⎥ ⎢ 0 1 0⎥ ⎣ ⎣ ⎦ ⎦ The following theorem establishes a relationship between the Seifert matrices formed by isotopic links. Theorem 4.7.20 Assume that . L1 and . L2 are isotopic links and .F1 and .F2 are their Seifert surfaces. If . A1 and . A2 are their Seifert matrices computed in some basis . B1 and . B2 , then . A1 is S-equivalent to . A2 . This was first established for knots embedded in .R3 by Murasugi in [Mura1]. Later, J. Levine established it for embeddings in higher dimensions [Lev1]. This theorem can be used to find a relation between the Seifert matrix and the Alexander polynomial. It is explained from a historical context below:

88

4 Goeritz and Seifert Matrices

Conway noticed that the Alexander polynomial can be normalized to satisfy the (Conway) skein relation. He did not give details except writing in [Con1], page 339: We have not found a satisfactory explanation of these identities [various skein relations], although we have verified them by reference to a “normalized” form of the “L-matrix” definition of the Alexander polynomial, obtained by associating the rows and columns in a natural way. It was formalized by Giller [Gi] and Kauffman [Kau1]. Exercise 4.7.21 Let A be a Seifert matrix of an oriented link L showing that the function: ( ) √ 1 T . det − t A + √ A t does not depend on the choice of a Seifert surface and its Seifert matrix. In fact, it is equal to the Alexander-Conway polynomial .Δ L (t). Hint: Use Theorem 4.7.20 describing S-equivalence of Seifert matrices (see, e.g., [Prz21]). This relationship between Seifert matrices helps establish a link invariant called the Tristram-Levine signature.

4.8 Tristram-Levine Signature We now discuss the Tristram-Levine signature—a generalization of the classical (Trotter-Murasugi) signature (see Definition 4.4.1 and [Gor, Lev2, PT2, Tri]). Moreover, in [Con2] there is a recent survey. Definition 4.8.1 A Hermitian form is a map . h : Cn × Cn → C which satisfies the following for any .a, b, c ∈ Cn : 1. . h(a + b, c) = h(a, c) + h(b, c). 2. For any .λ ∈ Cn , . h(λa, b) = λh(a, b). 3. . h(a, b) = h(b, a). The matrix H of a Hermitian form in any basis is called a Hermitian matrix (i.e., T . H = H ). A Hermitian form has a basis in which the matrix is diagonal with .1, −1, or 0 entries. Let .n1, n−1, and .n0 denote the number of .1' s, −1' s, and .0' s, respectively. These numbers form a complete invariant of a Hermitian form by Sylvester’s law of inertia. The nullity of the form is defined as .n0 . The signature of the form is given

4.8 Tristram-Levine Signature

89

by .σ = n1 − n−1 . If we count eigenvalues of h (with multiplicities), then .n1 is the number of positive eigenvalues of H and .n−1 is the number of negative eigenvalues. Definition 4.8.2 ([Tri, Lev2]) Let . AL be a Seifert matrix of a link L. For each complex number .ξ (.ξ = 1), consider the Hermitian matrix .HL (ξ) = (1 − ξ)AL + (1 − ξ)ATL . The signature of this matrix is called the Tristram-Levine signature of the link L. If the parameter .ξ is considered, we denote the signature by .σL (ξ); if we consider .φ = 1 − ξ as a parameter, we use the notation .σφ (L). The classical signature .σ satisfies .σ(L) = σ1 (L)σL (0) = σL (−1). Also, by an established convention, we write .σL (1) = 0 (see Remark 4.8.4). We now verify that the Tristram-Levine signature is a well-defined link invariant. A simple computation shows that if we replace the matrix A with another S-equivalent matrix, then .σ(HL (ξ)) remains the same. We establish the following identity: Lemma 4.8.3 [ ]) [ ] 00 01 ¯ = 0. + (1 − ξ) .σ (1 − ξ) 10 00 (

Proof Observe: .

[ Let . B =

]) ([ [ ]) ( [ ] 0 1 − ξ¯ ¯ 0 1 + (1 − ξ) 0 0 = σ . σ (1 − ξ) 1−ξ 0 10 00

] 0 1 − ξ¯ . We compute the eigenvalues of . B − λI: 1−ξ 0 ]) −λ 1 − ξ¯ . det(B − λI) = det 1 − ξ −λ ¯ − ξ). = λ2 − (1 − ξ)(1 ([

/ Setting this equal to 0, we obtain the eigenvalues .λ = ± 1 − ξ¯ − ξ + |ξ | 2 . If .ξ is complex, then .λ is complex and contributes no positive or negative eigenvalues then .n1 = 1 and .n−1 = 1. In either case, to (the signature. [ ] If .ξ is real [ valued, ]) 0 1 0 0 ¯ .σ (1 − ξ) + (1 − ξ) = 0 as desired. o 00 10 Remark 4.8.4 The signature of a Hermitian matrix is unchanged when the matrix is multiplied by a positive number.12 We can (and will) often assume that the .ξ in .σL (ξ) and the .ψ in .σψ (L) are of unit length. With these assumptions, we have Tristram-Levine signature functions, .σL (ξ), σψ (L) : S 1 → Z. .σL (ξ) is the signature function tabulated in [CL], and .σψ (L) is used in Example 4.8.6. .S 1 will be usually matrix H is Hermitian similar to .λH for any real positive number .λ; .λH = √ The Hermitian √ ( λI d)H( λI d)

12

90

4 Goeritz and Seifert Matrices

parameterized by .arg(ψ) ∈ [−π, π].13 Generally, we have .σL (ξ) = σ1−ξ (L), but when restricted to the unit circle, we have to write .σL (ξ) = σ (1−ξ ) (L). Notice that for .ψ = . .

1−ξ |1−ξ | ,

(1−ξ)(1−ξ) ¯ (1−ξ)(1− ξ)

we have .ψ 2 =

|1−ξ |

=

1−ξ 1− ξ¯

= −ξ (and .(iψ)2 = ξ). Therefore,

σψ (L) = σL ((iψ)2 ) = σL (ξ), for .Re(ψ) ≥ 0. By Corollary 4.8.5, we see that σi (L) = 0, which justifies the convention14 that .σL (1) = 0.

Corollary 4.8.5 ([Prz2]) 1. For any .t 2k -move and .Re(1 − ξ) ≥ 0 (i.e., . |arg(ψ)| ≤ π/2), we have: 0 ≤ σt¯2k (L) (ξ) − σL (ξ) ≤ 2.

.

In particular ([PT2]), for .Re(1 − ξ) ≥ 0, we have .−2 ≤ σL+ (ξ) − σL− (ξ) ≤ 0. 2. Furthermore, for .ξ and k, we have:

.

For a proof, we refer the reader to [Prz2] and [PT2]. Example 4.8.6 We now calculate the Tristram-Levine signature for the right-handed trefoil—.3¯ 1 . Recall the Seifert matrix . A3¯ 1 for the right-handed trefoil was calculated in Example 4.7.14 to be: [ ] −1 0 . A3 . ¯1 = 1 −1 Let .ξ be a complex number such that . ||ξ || = 1. We begin by computing the matrix: ] ] [ [ −1 1 ∗ T ∗ −1 0 + (1 − ξ) .(1 − ξ )A3 ¯ 1 + (1 − ξ)A3¯ = (1 − ξ ) 1 0 −1 1 −1 [ ] ∗ −2 + ξ + ξ 1−ξ = . 1 + ξ ∗ −2 + ξ ∗ + ξ In [CL], .S 1 is parametrized by . argξ π . In the literature on the Tristram-Levine signature of knots, the most often used normalization ¯ L is to take . |ξ | = 1(ξ = 1). When one writes the of the Hermitian matrix .(1 − ξ)A L + (1 − ξ)A function .σK (ξ), then the usual assumption about the parameter . ξ is that it is on the unit circle. ¯ L + (1 − ξ)A L ) = det((ξ − 1)( 1− ξ¯ A− AT )) = det((ξ − 1)( ξ¯ A− AT )) := Then one has .det((1 − ξ)A ξ −1 ¯ where .= denotes equality up to .±t i , [Gor]. When dealing with links, we found it (ξ − 1) n Δ( ξ), more convenient (see [PT2, Prz4]) to consider .ψ = 1 − ξ and to assume that . |ψ | = 1. Then we have ¯ A + ψ AT )) = det(i ψ¯ A − iψ AT ) = Ω(i ψ) ¯ = Ω(iψ) = ∇(−i(ψ¯ + ψ)). Here, .Ω denotes the .det(i(ψ potential function. Therefore, for any knot .σψ (K) = σK ((iψ)2 ) = σK (ξ). 13 14

4.9 Exercises

91

We compute the characteristic equation and take roots to find the eigenvalues. We compute the signature .σ = n1 − n−1 , where .n1 and .n−1 denote the number of positive and negative eigenvalues, respectively. We find the signature depending on the size of .Re(1 − ξ) in the following way: ⎧ −2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −1 ⎪ 0 .σ3 = ¯1 ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ 2 ⎩

if Re(1 − ξ) > 1/2 if Re(1 − ξ) = 1/2 if −1/2 < Re(1 − ξ) < 1/2 if Re(1 − ξ) = −1/2 if Re(1 − ξ) < −1/2.

This lecture has established many foundational definitions for two key matrix invariants in knot theory: the Goeritz matrix which relies on coloring of the surface components given by a link diagram and the Seifert matrix which relies on the construction of a special surface from the knot. Although these matrices are not invariants themselves, the formation of the matrices allowed for the construction of two key invariants: the determinant of the link and the classical signature in the case of the Goeritz matrix and the Alexander polynomial and the Tristram-Levine signature for the Seifert matrix. In the coming lectures, we continue our discussion of link invariants. However, we depart from the matrix-type invariants and instead establish polynomials that are associated with each knot. See Lectures 5 and 6 for the skein relation approach to the polynomial invariants.

4.9 Exercises Exercise 4.9.1 Find the determinants and signatures of knots 818 and 819 ; see tables in Appendix C. ] [ −1 0 ¯ . Exercise 4.9.2 Show that the Seifert matrix of the right-handed trefoil 31 is −1 −1 Exercise 4.9.3 Show that any oriented link has a special diagram, that is, a diagram in which Seifert circles are not nested. Exercise 4.9.4 Prove that an oriented diagram of a link is special if and only if all crossings are of type I for some checkerboard coloring of the plane. Exercise 4.9.5 ([Ban]) Show that an alternating diagram represents the trivial knot if and only if all its crossings are nugatory.

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4 Goeritz and Seifert Matrices

This observation was generalized to almost alternating diagrams15 by Tatsuya Tsukamoto [Tsu]. That is, an easy algorithm was given to determine whether an almost alternating diagram represents the trivial knot. Exercise 4.9.6 Show that lk(J, K) = lk(K, J) = −lk(−K, J) where −K denotes the knot K with reversed orientation. Exercise 4.9.7 Show that the Tristram-Levine signature for the figure-eight knot is equal to 0. Exercise 4.9.8 Show that the Tristram-Levine signature of an amphicheiral link is equal to 0. √ Exercise 4.9.9 Assume that the Alexander-Conway polynomial at t0 = −ψ 2 ( t0 = −iψ) is different from zero. Then: i σψ (L) =

.

Δ L (t0 ) , |Δ L (t0 )|

√ where Δ L (t0 ) is the Alexander-Conway polynomial and t0 = −ψ 2 ( t0 = −iψ). In particular, the Tristram-Levine signature is determined modulo 4 by the appropriate value of the Alexander-Conway polynomial; see [Prz21] for details.

15 A diagram is called almost alternating if it differs from alternating by exactly one crossing; see [Ada]

Lecture 5 The Jones Polynomial and Kauffman Bracket Polynomial

In May of 1984, Vaughan Jones defined his link polynomial, and later in the summer of 1985, Louis H. Kauffman described his bracket polynomial. In this lecture, we describe basic properties of these polynomials including mysterious relations with Fox 3-coloring. We also discuss Montesinos-Nakanishi 3-move conjecture and its solution using the Burnside group of link. We end by discussing the Nakanishi 4-move conjecture, from 1979.

5.1 Introduction In July of 1984, the first author attended a conference in Durham, England, where Hugh Morton brought the news that Vaughan Jones had a totally novel polynomial knot invariant which, unlike the Alexander polynomial, often distinguishes mirror images. Morton brought with him the letter written by Jones to Joan Birman from May 1984 [Jon2], where the basic construction of this polynomial (using traces of von Neumann algebras) was described. This letter, with subsequent talks by Jones, made a lasting impression on the topological community. The invariant looked totally unlike everything known until then, for example, there was no obvious relation with the fundamental group of a link complement. The polynomial was a breakthrough in knot theory and led to the rapid growth of this discipline. Some of the new developments are discussed in these lectures. In particular, soon after the discovery of the Jones polynomial, the HOMFLYPT polynomial was constructed. This directed Pawel Traczyk and the first author toward knot theory. The HOMFLYPT polynomial will be discussed in Lecture 6. We now define the Jones polynomial via the skein relation noted by Jones and prove its existence using an elementary construction given by Louis Kauffman during the summer of 1985. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_5

93

94

5 The Jones Polynomial and Kauffman Bracket Polynomial

5.2 The Jones Polynomial via Skein Relations As the story goes, Jones had already noted the skein relation in May 1984; however, this was not included in his letter to Birman. Jones’ definition, using the recursive skein relation, is as follows: (i) Initial data: .VO = 1, where .O is the trivial knot, √ (ii) . 1t VD+ (t) − tVD− (t) = ( t − √1t )VD0 (t). The diagrams of oriented links . D+ , . D− , and . D0 are different only in small disks as pictured in Fig. 5.1.

D+

D-

D0

Fig. 5.1: Conway’s skein triple We first show that these rules allow computing the Jones polynomial for any link diagram, that is, we show the uniqueness of the Jones polynomial. Definition 5.2.1 Let D be an oriented link diagram of n components with base points b = (b1, . . . , bn ) chosen on every component (b is disjoint from the crossings of D). We say that D is a descending diagram with respect to b if the following property holds: we travel along the link according to the orientation of the link diagram starting at .b1 . After the first component is traveled, we start from .b2 , etc. The property is that if we approach a crossing for the first time we travel along the bridge.

.

Informally, we can say that a descending diagram is obtained if we draw the diagram on the blackboard without erasing. Exercise 5.2.2 Show that any descending diagram represents a trivial link. Hint: With respect to the height function, each component has one maximum and one minimum, and the components are placed in layers one above another. Proposition 5.2.3 The following algorithm will always compute the Jones polynomial of an oriented diagram D. Choose base points .b = (b1, . . . , bn ) and move along the diagram as in Definition 5.2. Count the number of “bad” crossings, that is, crossings which are not over-passed first when traveling. The complexity of .(D, b) is the pair .(c, d) where c is the number of crossings of D and d the number of “bad” crossings. We order this pair lexicographically. For .d = 0 we deal with the descending diagram, so by Exercise 5.2.4(i), we know its Jones polynomial. Now let v be the

5.2 The Jones Polynomial via Skein Relations

95

first “bad” crossing. We then use the Jones skein relation at v and notice that other diagrams involved in the relation have smaller complexity than the original diagram v has less “bad” crossings). Thus, we D (. D0v has less crossings than D and . D−sgn(v) have an algorithm by induction on the complexity. We will prove the existence of the Jones polynomial in Sect. 5.4 using ideas by Kauffman. We now propose a few basic but important exercises. Exercise 5.2.4 (i) Show that adding the trivial component to a link L results in the formula: √ 1 VLuO (t) = (− t − √ )VL (t). t

.

In particular, show that for a trivial link of n components, .Tn , we have: √ 1 VTn (t) = (− t − √ )n−1 . t

.

Hint: Consider the skein triple . L+, L−, L0 from Fig. 5.2.

L+

L

L0

Fig. 5.2: Skein triple to compute the Jones polynomial of . L u O (ii) Let . D¯ denote the mirror image of the oriented diagram D. Then: VD¯ (t) = VD (t −1 ).

.

Notice that the mirror image is well-defined for links, not only diagrams, because all mirrors (i.e., 2-plains) are isotopic in .R3 . (iii) Compute the Jones polynomial of the positive (right-handed) Hopf link (

) and trefoil (

) from the definition (the fact that .V3¯ 1 is not sym-

metric was, for Jones, the first hint that he was dealing with a new link invariant).

(iv) Compute the Jones polynomial of the figure-eight knot (

).

96

5 The Jones Polynomial and Kauffman Bracket Polynomial

(v) Show that the Jones polynomial of the disjoint sum . L1 u L2 of links . L1 and . L2 satisfies: √ 1 .VL1 uL2 (t) = (− t − √ )VL1 (t)VL2 (t). t (vi) Show that the Jones polynomial of the connected sum1 . L1 #L2 of links . L1 and . L2 satisfies: .VL1 #L2 (t) = VL1 (t)VL2 (t). Exercise 5.2.5 Let .−L be the oriented link obtained from oriented link L by changing its orientation. Show that .V−L (t) = VL (t). Remark 5.2.6 L is called non-invertible if L is different than .−L. The smallest noninvertible knot is .817 ; see Appendix C. It is the only non-invertible knot of 8 or less crossings. There are 2 such knots of 9 crossings, 33 of 10 crossings, and 187 of 11 crossings. Starting from 12 crossings, there are more non-invertible than invertible crossing knots (for 12-crossings: 1144 vs. 1032.).2 Exercise 5.2.7 (Jones’ Observation) Show that for any oriented knot K, .VK (t) − 1 is divisible by .(t − 1)(t 3 − 1). Hint: Using Jones’ skein relation, first show that for an oriented link of n components, .VL (t) − VTn (t) is divisible by .t 3 − 1. For example, for the right-handed trefoil knot .3¯ 1 , we have .V3¯ 1 − 1 = (1 − t)(t 3 − 1). Jones proposed the following normalized version of his polynomial for knots: WK (t) =

.

VK − 1 , (1 − t)(t 3 − 1)

so that .W3¯ 1 (t) = 1. Exercise 5.2.8 Find .W41 (t). Conjecture 5.2.9 (Jones Conjecture [Jon9]) The only knot with the trivial Jones polynomial is the trivial knot. 1 A diagram .D1 #D2 is a connected sum of diagrams .D1 and .D2 if there is a simple closed curve cutting .D1 #D2 in exactly two points and 1-tangles obtained by cutting .D1 #D2 by the curve have . D1 and . D2 as their closures. A link . L1 #L2 is a connected sum of links . L1 and . L2 if there is a diagram of . L1 #L2 which is a connected sum of diagrams of . L1 an . L2 . Connected sum is unique for oriented knots [Schu1]; however, in general, it may be not unique for links. More precisely, the result may depend on the components of the link used in the connected sum and on the orientation of the link. Compare with Definition 4.6.11. 2 Trotter was the first to find a non-invertible knot. He proved that if . p, q, r are odd integers such that . |p |, |q |, and . |r | are distinct and greater than 1, then the pretzel knot . K(p, q, r) is non-invertible [Tro].

5.3 The Kauffman Bracket Polynomial

97

The conjecture has been confirmed for several families of knots, including alternating, adequate knots, and knots up to 24 crossings [LT, Yam2, TuSi]. The analogous conjecture for Khovanov homology was solved in [KrMr] (see Lecture 19). Jones computed his polynomial for torus knots of type .(m, n) (see Lecture 6, e.g., Fig. 6.6, for the definition of .T (m, n)). His closed formula is very nice but not so easy to prove, so it makes a rather difficult exercise. Exercise 5.2.10 ([Jon4]) Show that for the torus knot .T (m, n) we have: VT (m,n) (t) = t m(n−1)/2

.

t (k+1)/2 − t (−k−1)/2 − t m (t (k−1)/2 − t (1−k)/2 ) . t − t −1

Remark 5.2.11 The formula can be generalized to torus links in an annulus (formula in KBSM of .Ann × I); see [Prz17]. A simple proof was given in [FrGe] using the product-to-sum formula in the KBSM of the Cartesian product of the torus and the interval, .T 2 × I. Kauffman bracket skein modules are discussed in Lecture 12.

5.3 The Kauffman Bracket Polynomial The Kauffman bracket polynomial . was defined by Kauffman in the summer of 1985 independent of the Jones polynomial, [Kau6]. , and can be He was investigating the possibility that three diagrams linked by a linear relation leading to a link invariant. Only later did he realize that he had constructed a variant of the Jones polynomial. Initially, Kauffman considered arbitrary variables (so he had chosen A and B since they are first letters of the alphabet), that is, for an unoriented diagram of a link, he considered the skein equation with initial conditions . = d n−1 , where .Tn is the trivial diagram of n components. Thus, .< D > is a polynomial in three variables A, B, and d (i.e., .< D > A,B,d ∈ Z[A, B, d]). We leave the fact that this polynomial is well-defined for any diagram as an exercise. Exercise 5.3.1 Show that . is a well-defined polynomial for an unoriented link diagram. Exercise 5.3.2 Show that the three-variable polynomial .< D > A,B,d computed for reduced alternating diagrams is a link invariant.

98

5 The Jones Polynomial and Kauffman Bracket Polynomial

5.4 The Jones Polynomial from the Kauffman Bracket and Their Relation to Fox 3-Coloring We analyze (after Kauffman) how the three-variable Kauffman polynomial behaves under Reidemeister moves and show, in particular, that the second Reidemeister move suggests the substitution . B = A−1 and .d = −A2 − A−2 :

.

The natural solution for is to put . B = A−1 and .d = −A2 − A−2 . Invariance under the third Reidemeister move follows almost automatically by comparing the following two equations representing the sides of the third Reidemeister move:

.

and

.

Both formulas give the same result as long as the bracket is preserved by the second Reidemeister move. Thus, we have proved that the Kauffman bracket of unoriented links is preserved by the second and third Reidemeister moves. In this case, we say that the bracket polynomial is an invariant of regular isotopy. Another, equivalent, interpretation of the Kauffman bracket polynomial is by considering framed links, that is, that we study embedded annuli not circles. In this interpretation, the first Reidemeister move is changing a framing of the link. This point of view will be used in Lecture 12 where the Kauffman bracket polynomial is generalized to the Kauffman bracket skein module of any 3-manifold. Notice that for the first Reidemeister move, we get: .

= −A3 , = −A−3 .

We can summarize our calculations as follows:

5.4 The Jones Polynomial from the Kauffman Bracket and Their Relation to Fox 3-Coloring

99

Theorem 5.4.1 ([Kau6]) Let D be an unoriented diagram of a link. Then the Kauffman bracket polynomial . ∈ Z[A±1 ] described by the following properties: (i) . = 1 (ii) . = −(A2 + A−2 ) (iii) is an invariant of the second and the third Reidemeister moves. Under the first positive Reidemeister move, . = −A3 , and under the negative move, −3 . . = −A We should note that the Kauffman bracket given in the previous theorem is a version of the Kauffman bracket, normalized to be 1 for the unknot, and called the reduced Kauffman bracket. The version normalized to be 1 for the empty link, denoted by .[D], is called the unreduced Kauffman bracket (thus, we have .[D] = (−A2 − A−2 )). We will show how to balance the first Reidemeister move by the writhe number of an oriented diagram and obtain the version of the Jones polynomial for oriented links. First, we show how to combine the Kauffman bracket relations so that they give a relation similar to the Jones skein relation. Consider the Kauffman bracket skein relation: .

and its variant rotated by 90 degrees: .

We can now eliminate from the equations of generality, we eliminate to get:

or

, and without loss

.

We will now balance the bracket so that the equation above gives the Jones skein relation (it is already close to it). → − Definition 5.4.2 Recall from Lecture 2 that for an oriented diagram . D, we define the writhe (or Tait) number of the diagram as the sum of the signs of all its crossings, E → − sgn(c). that is, .w( D) = c ∈cr(D)

Exercise 5.4.3 Show that: .

- = w( D) - + 1, w(R1− ( D)) - = w( D) - − 1, w(R1+ ( D))

100

5 The Jones Polynomial and Kauffman Bracket Polynomial

- is an invariant of regular isotopy (i.e., .w( D) - is invariant under and furthermore .w( D) the second and the third Reidemeister moves). - to balance .< D >. Therefore, by Exercise 5.4.3, we can use .w( D) - be any oriented diagram obtained from D. We For an unoriented diagram D, let . D check that the polynomial: .

fD- (A) = (−A3 )−w(D) < D > -

is invariant under all three Reidemeister moves. Without loss of generality, the strings are oriented horizontally, so we can write: .

Using the polynomial . fD- , we get: .

Thus, after reduction we get:

.

After substituting .t = A−4 our equation changes to the skein relation of the Jones polynomial: .

Therefore, for .t = A−4 , we have . fD- (A) = VD- (t). Thus, the existence of the Jones polynomial is proven. If we take the sum of

and

, then we obtain the equation:

.

which is a certain substitution of the 2-variable Kauffman polynomial (see Lecture 6). If we take the difference of then we obtain the equation: and .

which is a certain substitution of the Dubrovnik polynomial (a version of the 2variable Kauffman polynomial described in Lecture 6). Theorem 5.4.4 ([Prz13]) The following identity connects Fox 3-colorings and the Jones polynomial for any oriented link L:

5.4 The Jones Polynomial from the Kauffman Bracket and Their Relation to Fox 3-Coloring 101

| π i |2 col3 (L) = 3|VL (e 3 )| .

.

The simplest proof of this result is given by studying the homology of the double branched cover of .S 3 , branched over a link. However, the formula was found differently: 3-moves preserve both sides of the equation, so as long as a link can be reduced to an unlink by 3-moves, the theorem easily holds. This is related to the Montesinos-Nakanishi 3-move conjecture (see Conjecture 5.7.1). We challenge the readers to find a full elementary proof of the theorem. We can modify a link diagram by n-twists (called n-move) as illustrated in Fig. 5.3. The following proposition expresses . as a function of . and . . Proposition 5.4.5 We have the following formula for the result of an n-move: .

1 + (−1)n−1 A4n

1 + A4 = An + (−1)n−1 A−3n+2 [n]−A4 .

= An + (−1)n−1 A−3n+2

In particular, for . A4n = (−1)n , . A4 = −1, we have . = An . Proof Use the Kauffman bracket skein relation to get . = A+ A−1 (−A)−3(n−1) and iterate (it until we get the formula of the ) proposition: n n−1 A−3n+2 1 − A4 + A8 + . . . + (−1)n−1 A4n−4 . . = A + (−1) o ∞ .

n− move

D0

Dn

n right handed half twists

Fig. 5.3: n-moves on a link diagram (. D0 → Dn ) As a corollary, we propose the following exercise: Exercise 5.4.6 (a) Find a closed formula for the Kauffman bracket polynomial and the Jones polynomial, of the torus link of type .(2, n); see Fig. 5.4. (b) Find a closed formula for the Kauffman bracket polynomial and the Jones polynomial, of the twist knot; see Fig. 5.5. Hint: (a) Use Proposition 5.4.5 and the fact that in our case . = −A2 − A−2 and . = 1 Exercise 5.4.7 Find a formula for the Kauffman bracket polynomial of a pretzel link P(n1, n2, . . . , nk ). Figure 5.6 illustrates the pretzel knot .P5,7,−3 .

.

102

5 The Jones Polynomial and Kauffman Bracket Polynomial

Fig. 5.4: The torus link diagram of type .(2, n)

Fig. 5.5: The twist knot diagram of .n + 2 crossings

Fig. 5.6: The pretzel knot .P5,7,−3

5.5 The First Tait Conjecture from the Kauffman Bracket Polynomial One of the first spectacular applications of the Jones polynomial (via the Kauffman bracket) was the proof by Kauffman [Kau3], Murasugi [Mura2, Mura3], and Thistlethwaite [Thi2] of the first Tait conjecture for alternating knots (see Lecture 2). In this section, we present the proofs of the first and the second Tait conjectures. The

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proof of the third requires Menasco’s work in [MeTh] on incompressible surfaces and is outside of the scope of this book. We formulate, and prove here, a slight generalization of the first Tait conjecture: Theorem 5.5.1 (The First Tait Conjecture) (i) A reduced alternating diagram of a given link has a minimal number of crossings among all diagrams representing the link. In particular, two reduced alternating diagrams of the same link have the same number of crossings. (ii) If we assume additionally that our link is prime and non-split (i.e., it is not a split link or a connected sum of links), then any nonalternating diagram of the link has a non-minimal number of crossings. The proof we present is relatively simple as for a 100-year-old conjecture. Below we list the main tools and steps used; some of them are explained in detail and some are left as exercises for the reader. (1) Kauffman’s state sum formula for the Kauffman bracket polynomial. (2) Extreme coefficients of the Kauffman bracket polynomial of A- and B- adequate diagrams. (3) Adequate (respectively, A- and B-adequate) diagrams. (4) Reduced alternating diagrams are adequate diagrams. (5) Dual states and Wu’s lemma. (6) Turaev surfaces. Definition 5.5.2 ([Kau3]) The Kauffman state of a given diagram D is any function s : cr(D) → { A, B}. The set of such functions is denoted by .KS. The interpretation of a state is given in Fig. 5.7; in particular, . Ds is the diagram (system of circles) obtained from D by applying markers of s at every crossing of D and . |Ds | denotes the number of circles in . Ds .

.

Fig. 5.7: Kauffman state function

Proposition 5.5.3 (The Kauffman Bracket State Sum Formula) .

=

E s ∈KS

Aσ(s) (−A2 − A−2 ) |Ds |−1,

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where .σ(s) is the number of A-markers in s minus the number of B-markers in s (i.e., .σ(s) = |s−1 (A)| − |s−1 (B)|). Proof We use the Kauffman bracket relation simultaneously to all crossings and o then apply initial condition to . Ds for any s. We use the Kauffman bracket state sum formula to find the states providing, potentially, the highest (and lowest) power of A in the Kauffman bracket polynomial. We concentrate first on the coefficient of the highest power of A. From the formula, we see that .s A provides the monomial .(−1)Ds A Acr(D)+2 |Ds A |−2 in the state sum formula for . . We will show that no state brings a higher power of A and we give criteria for other states to bring the same power. Lemma 5.5.4 (1) Consider two Kauffman states s and .s ' which differ at exactly one crossing, say v, at which .s(v) = A and .s '(v) = B; in particular, .σ(s ') = σ(s) − 2 and ' . |Ds' | = |Ds | ± 1. Then the maximal power of A provided by . s to . is either equal to or smaller by 4 than the maximal power provided by s. Specifically, either . |Ds' | = |Ds | − 1 (we say that . Ds has a mixed touching at v) and the power of A is going down by 4 or . |Ds' | = |Ds | + 1 (we say that . Ds has a self-touching at v) and the maximal power of A is preserved. (2) The maximal power of A in . is bounded by .cr(D) + 2|Ds A | − 2, that is, .maxdeg ≤ cr(D) + 2|Ds A | − 2. (3) If . Ds A has no self-touchings, then no other state brings the monomial of exponent .cr(D) + 2|Ds A | − 2, and .max deg = cr(D) + 2|Ds A | − 2. (4) A Kauffman state s contributes to . a term with A-exponent .cr(D)+2|Ds A | −2 if and only if there is a sequence of states, .s A = s1, s2, . . . , sk = s such that .si+1 differs from .si at exactly one crossing, say .vi and .si (vi ) = A, .si+1 (vi ) = B, and . |Dsi+1 | = |Dsi | + 1, that is, . Dsi has a self-touching at .vi . Proof Part (1) follows directly from the Kauffman bracket state sum formula. Parts (2), (3), and (4) can be proved simultaneously. Let s be an arbitrary Kauffman state and .s A = s1, s2, . . . , sk = s the sequence such that .si+1 differs from .si at exactly one crossing, say .vi and .si (vi ) = A, .si+1 (vi ) = B. For any i, we have . |Dsi+1 | = |Ds |±1. We use (1) to conclude (2), (3), and (4). o Condition (3) motivated Lickorish and Thistlethwaite [LT] to introduce the notion of adequate diagrams.

5.5.1 Adequate Links and Diagrams Definition 5.5.5 (Adequate Links and Diagrams) Let s A (respectively, sB ) be a Kauffman state of a diagram D such that s A(v) = A

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(respectively, sB (v) = B) for any crossing v of D. Let s be a Kauffman state which differs from s A (respectively, sB ) in exactly one crossing, say v. The diagram D is called A-adequate (respectively, B-adequate) if Ds has one less component from Ds A (respectively, from DsB ) for any crossing v; in other words, components of Ds A (respectively, DsB ) have no self-touching. Equivalently, the Tait graph of D has no loop (respectively, its dual has no loop). Thus, we have |Ds | < |Ds A | (respectively, |Ds | < |DsB |). A diagram is called adequate if it is A- and B- adequate, and the link is adequate (respectively, A- or B-adequate) if it has a diagram which is adequate (respectively, A- or B-adequate); compare Lecture 20. We formulate the following important corollary to Lemma 5.5.4 using the language of adequate diagrams. Corollary 5.5.6 Let D be a link diagram, and then: (1) maxdeg ≤ cr(D) + 2|Ds A | − 2 and for A-adequate D the equality holds. (2) mindeg ≤ −cr(D) − 2|DsB | + 2 and for B-adequate D the equality holds. (3) span A ≤ 2cr(D) + 2(|Ds A | + |DsB |) − 4 and for an adequate D the equality holds. Proof (1) Follows directly from Lemma 5.5.4. ¯ (2) Follows from (1) by replacing D with its mirror image D. (3) Follows directly from (1) and (2).

o

Remark 5.5.7 We can modify the definition of adequacy, which is geometric in flavor, into algebraic definition which uses the Kauffman bracket polynomial: A link diagram D is Kauffman bracket (succinctly, bracket) A- (respectively, B-) adequate if maxdeg = cr(D) + 2|Ds A | − 2 (respectively, mindeg = −cr(D) − 2|DsB | + 2). D is bracket adequate if it is simultaneously A- and B- bracket adequate. A link L is bracket adequate (respectively, A- or B- bracket adequate) if there is a diagram D L , representing L which is bracket adequate (respectively, A- or B- bracket adequate). One can give a similar definition using Khovanov homology, but we leave the study of Khovanov adequate diagrams to the readers after they read Lectures 19 and 20.3 Exercise 5.5.8 Show that for a bracket adequate diagram, we have: span A = 2cr(D) + 2(|Ds A | + |DsB |) − 4.

.

3 A link diagram D is Khovanov A- (respectively, B-) adequate if Ha, b (D) is nontrivial for b = cr(D) + 2|D s A | (respectively, b = −cr(D) − 2|D s B |). D is Khovanov adequate if it is simultaneously A and B Khovanov adequate. A link L is Khovanov adequate (respectively, A- or B- Khovanov adequate) if there is a diagram D L , representing L which is Khovanov adequate (respectively, A- or B-Khovanov adequate); see [PrSi].

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5.5.2 Alternating Links Are Adequate The main reason to introduce adequate diagrams was because they generalize and share some properties of alternating diagrams. Exercise 5.5.9 Show that: (i) Reduced alternating diagrams are adequate. (ii) An alternating diagram is A-adequate if and only if all its nugatory crossings are positive. (iii) An alternating diagram is B-adequate if and only if all its nugatory crossings are negative. Hint: (i) Consider a checkerboard coloring of .R2 − D and use Listing’s observation that connected alternating diagram corners of crossings of a given color have all A or all B colors (compare Exercise 4.5.4). Then notice that the alternating condition prevents “self-touching” of regions. For (ii) and (iii), consider Fig. 5.8.

s(v)=A

s(v)=B

Fig. 5.8: A- and B-nugatory crossings

5.5.3 Wu’s Lemma One can finish the proof of the first Tait conjecture by analyzing the sum . |Ds A |+|DsB | and proving the following lemma used in a different form by all researchers who proved the conjecture. Lemma 5.5.10 (On Dual States [Kau3, Wu1]) Let D be a connected diagram of a link. Then we have the following: (1) If s is a Kauffman state of D, let .s* denote the dual state, that is, .s−1 (A) = (s*)−1 (B). Then . |Ds* |+|Ds | ≤ cr(D)+2, in particular . |Ds A |+|DsB | ≤ cr(D)+2. (2) The equality holds if and only if D is a reduced alternating diagram or a connected sum of such diagrams.

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Hint: A simple way to show Lemma 5.5.10 is to translate it to the language of signed graphs (following Tait as in Fig. 2.9). The first Tait conjecture follows directly from Corollary 5.5.6 and Wu’s lemma. In particular, we observe that for a non-split alternating link, .cr(L) = 14 span A = spant VL (t).

5.5.4 Turaev’s Construction of TF(D) In fact, we can interpret . |Ds A | + |DsB | − cr(D) as the Euler characteristic of a certain closed oriented surface, first considered by Vladimir Turaev [Tur1] and named by Oliver Dasbach, Turaev surface in [DFKLS]. From this, we immediately get that 2 . |Ds A | + |Ds B | − cr(D) ≤ 2 and the equality holds if and only if the surface is . S . The genus of the surfaces is called the Turaev genus of D and the surface is denoted by .TF(D). In our presentation, we follow the original definition of Turaev. Definition 5.5.11 (Turaev Surface) (1) Let D be a link diagram. The Turaev surface of D, denoted by .TF(D), is a closed surface defined as follows. Let . Dpr denote the 4-valent graph being the projection of D on .R2 . Further, let pr which is a plane surface. We modify .VD pr be the regular neighborhood of . D this surface to a new surface by half-twisting some ribbons corresponding to edges of . Dpr . Informally, we twist ribbons corresponding to edges connecting nonalternating crossings.4 It is precisely described in Fig. 5.9 (we perform the twist in such a way that . Ds A is always above . DsB ; however, it is not important for our considerations which direction we twist). In such a way, we obtain a surface with . Ds A u DsB as a boundary, so it has . |Ds A | + |Ds A | boundary components and the Euler characteristic is equal to .−cr(D). We denote this surface by .TF ∂ (D) and call it a Turaev surface with boundary. The closed surface .TF(D), known as the Turaev surface of D, is obtained from .TF ∂ (D) by capping off the boundary components. (2) Turaev genus of a link L, denoted by .g(TF(L)), is the minimal genus of the Turaev surfaces over all diagrams representing L. We note that the Turaev surface with boundary, .TF ∂ (D), is a cobordism between Ds A and . DsB .

.

Exercise 5.5.12 Show that the Turaev surface of a link diagram is orientable.

4 Turaev calls edges connecting alternating crossings “true edges” and edges connecting nonalternating crossings “wrong edges”; see Fig. 5.9.

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Surface along s−true edge

Surface along s−wrong edge Fig. 5.9: Construction of Turaev surfaces from alternating and nonalternating pieces

Turaev proved that .TS(D) = S 2 if and only if D is a connected reduced alternating diagram or a connected sum of such diagrams. The characterization of diagrams for which .TS(D) = T 2 is given in [AL, Kim].

5.5.5 Connected Sum of Alternating Links Is Alternating Tait initially thought that the connected sum of alternating knots (e.g., square knots as shown in Fig. 5.10) is not necessarily alternating. We think that he noticed his mistake and found a genuine nonalternating knot (today, it is denoted by .819 ; see Fig. 2.9); see [Tai3]. Exercise 5.5.13 Consider two alternating link diagrams . D1 and . D2 and their connected sum so that the resulting diagram of . D1 #D2 is not alternating. The standard diagram of the square knot (.3¯ 1 #31 ) is usually drawn using a nonalternating diagram (see Fig. 5.10). Show that there is another alternating diagram representing the same link as . D1 #D2 .

Fig. 5.10: The standard diagram of the square knot, .3¯ 1 #31 and its alternating diagram

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109

Hint: Note that the Tait graph .G(D1 #D2 ) is the wedge of a positive graph (.G(D1 )) and negative graph .G(D2 ). Take the dual graph (see Definition 10.4.1) to .G(D2 ) and take the diagram . D1 #D(G∗ (D2 )) (all edges in the graph are positive). We get an alternating diagram representing the same knot as . D1 #D2 .

5.5.6 The Second Tait Conjecture from the Jones Polynomial and Signature

Thistlethwaite used the Kauffman two-variable polynomial to prove the second Tait conjecture. Murasugi independently proved this conjecture in [Mura3] by using the Jones polynomial and the signature of links. Namely, he showed the following dependence between the Jones polynomial and the signature. Theorem 5.5.14 ([Mura3]) For any connected diagram of a link, L, we have: 1. .max VL (t) ≤ n+ (L) − + 12 σ(L) and 2. .min VL (t) ≥ −n− (L) − 12 σ(L), where .n+ (L) (respectively, .n− (L)) denotes the number of positive (respectively, negative) crossings of the diagram L. Both inequalities simultaneously become equalities if and only if L is an alternating diagram without nugatory crossings or it is a connected sum of such diagrams. We demonstrate, after Traczyk, the equality for alternating diagrams in Lecture 4 (also compare with [Prz21]). Corollary 5.5.15 If L is either an alternating diagram with no nugatory crossing or a connected sum of such diagrams, then the Tait number .Tait(D) satisfies the formula: Tait(D) = n+ (D) − n− (D) = max VD (t) + min VD (t) + σ(D).

.

The Tait number is also known as the writhe number .w(d). The second Tait conjecture follows from Corollary 5.5.15.

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5 The Jones Polynomial and Kauffman Bracket Polynomial

5.6 Mutations The simple but ingenious method of producing different links which are difficult to distinguish by known invariants is mutation invented before 1960 by J. H. Conway.5 We describe the Conway idea of tangles and mutation below:6 Consider a 2-tangle, L, that is a part of a link diagram placed in a disk, with four boundary points (two inputs and two outputs, if L is oriented); see Fig. 5.11.

(a) An unoriented tangle.

(b) An oriented tangle.

(c) An oriented tangle.

Fig. 5.11: Unoriented and oriented 2-tangles We perform a mutation of the link, of which L is a part, by rotating the tangle along the . x, y or z coordinate axis by the angle .π. Thus, we have three mutations .m x , my and .mz , respectively (see Fig. 5.12) Notice that together with the identity map, they form the group . D2 = Z2 ⊕ Z2 . We keep the part of the link outside the

mx my mz

Fig. 5.12: Mutations 5 Hugh Morton writes [Mor3]: Remarkably little of John Conway’s published work is on knot theory, considering his substantial influence on it. He had a really good feel for the geometry, particularly the diagrammatic representations, and a knack for extracting and codifying significant information. He was responsible for the terms tangle, skein and mutant, which have been widely used since his knot theory work dating from around 1960. Many of his ideas at that time were treated almost as a hobby and communicated to others either over coffee or in talks or seminars, only coming to be written in published form on a sporadic basis. 6 Conway recalls working out a good part of his theory of tangles while still a high school student [Alb].

5.7 Montesinos-Nakanishi 3-Move Conjecture

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tangle fixed and, if necessary, change the orientation of the tangle part so it agrees with the outside part of the link. Exercise 5.6.1 Show that the Kauffman bracket and the Jones polynomials are invariant under mutation. The mutation preserves not only the Jones but also HOMFLYPT and Kauffman polynomials (see Lecture 6). It also preserves the colored Jones polynomial and the volume of the (hyperbolic) complement of a link and the homeomorphic type of the branched double cover of .S 3 with the link as the branching set, among many other invariants. Figure 5.13 shows the mirror of the Conway knot .11n34 and the knot .11n42 which is the mirror image of the Kinoshita-Terasaka knot. The knot .11n42 is a mutant of the mirror image of the Conway knot; see [KAT].

(a) The 11

34

knot also known as the mirror of the Conway knot.

(b) The Kinoshita-Terasaka knot.

Fig. 5.13: Mutant knots

5.7 Montesinos-Nakanishi 3-Move Conjecture We found in Theorem 5.4.4 a connection between the number of Fox 3-colorings, col3 (L), and the Jones polynomial of L and mentioned its connection to 3-moves and Montesinos-Nakanishi 3-move conjecture. We will develop the topic in this subsection. First, how the connection was found? The first author noticed that the number of Fox n-colorings, .coln (D), is invariant under n-moves and studied the behavior of the Kauffman bracket polynomial (and the Jones polynomial) under nmoves (see Proposition 5.4.5) and noticed that both sides of the formula in Theorem 5.4.4 are preserved by 3-moves. Also the equality holds for trivial links. Thus, as far as a link can be reduced to a trivial link by a finite sequence of 3-moves, the Theorem 5.4.4 holds. On the other hand, he knew about the following conjecture, so Theorem 5.4.4 was a natural guess.

.

Conjecture 5.7.1 ([Kir1]) (Montesinos-Nakanishi 3-move conjecture) Every link can be reduced by 3-moves to a trivial link.

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Exercise 5.7.2 Show that torus links of type .(2, n) (including the trefoil knot) and twist knots (including the figure-eight knot) are 3-move equivalent to trivial links (i.e., they can be reduced to a trivial link by a finite sequence of 3-moves). Exercise 5.7.3 Show that trivial links of different number of components are not 3-move equivalent. Hint: Show, the fact mentioned before, that a 3-move preserves the number of Fox 3-colorings of a link. Yasutaka Nakanishi introduced the conjecture in 1981. José Montesinos analyzed 3-moves before, in connection with threefold dihedral branch coverings, and asked a related but different question. The conjecture was proved in many special cases (e.g., [Che]) but it was an open problem for over 20 years. In 2002, it was showed by Mieczyslaw Dabkowski and the first author that it does not hold in general. The following is the counterexample of 20 crossings [DP1, DP2, Prz24].

Fig. 5.14: 20-crossing link—the closure of the 5-string braid .(σ3 σ4−1 σ3 σ1−1 σ2 )4

To solve the Montesinos-Nakanishi 3-move conjecture, the authors of [DP1, DP2] introduced the concept of Burnside group of a link. The definition is as follows:

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Definition 5.7.4 (1) The core group, .Core(D), of a link diagram D is defined as follows: the generators of the group are indexed by the arcs of the diagram. The relations are given by the crossings of the diagram and they are of the form .ba−1 bc−1 , where .a, b, and c are the generators associated with the arcs of the crossing as in the figure below.

.

b

a

c

b

(2) The nth Burnside group of a link diagram . Bn (D) is defined as a quotient n n .Π(D)/(w ) where the .(w ) is the normal subgroup generated by all elements of n the type .w . Exercise 5.7.5 Show that the core group and nth Burnside groups are preserved by Reidemeister moves, so that they are link invariants. Exercise 5.7.6 Show that the nth Burnside group is preserved by an n-move. Exercise 5.7.7 Compute the core group and nth Burnside group for the trefoil knot and the figure-eight knot. Exercise 5.7.8 Show that the link illustrated in Fig. 5.14 and the trivial link of five components are not 3-move equivalent. Let us finish this lecture with two conjectures by Yasutaka Nakanishi from 1979 and by Akio Kawauchi from 1985, which are still open. Conjecture 5.7.9 ([Kir1, DP2]) (1) (Nakanishi 4-move conjecture) Every knot can be reduced by 4-moves to the trivial knot. (2) (Kawauchi 4-move conjecture) Every link of two components can be reduced by 4-moves to the trivial link of two components or to the Hopf link. Remark 5.7.10 The Nakanishi 4-move conjecture is proven for knots up to 12 crossings, alternating links up to 12 crossings, close 3-braids, and 2-algebraic links; see [Kir1, DJKS, Prz27].

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5.8 Exercises Exercise 5.8.1 Compute the Jones polynomial of the left-handed trefoil knot from the skein relation definition. Compare the answer from Exercise 5.2.4 (iii).

Exercise 5.8.2 Make an observation relating to the Jones polynomial of the figureeight knot and its mirror image.

Exercise 5.8.3 Use Exercise 5.2.4 (vi) to compute the Jones polynomial of the granny knot 31 #31 and the square knot 31 #3¯ 1 . Exercise 5.8.4 Compute the Jones polynomial and the Kauffman bracket of the Conway knot as seen in Fig. 5.13(a). Conclude what the Jones polynomial and Kauffman bracket of the Kinoshita-Terasaka knot (Fig. 5.13(b)) is equal to and explain why.

Exercise 5.8.5 Is the Conway knot chiral (i.e. non-amphicheiral)? Explain why or why not. Exercise 5.8.6 (1) Let D be a link diagram and v its self-crossing, that is, a crossing involving only one component of the link. Show that the sign of the crossing does not depend on the orientation of D. (2) The self-writhe, sw(D), of an oriented diagram D is defined E as the sum over all self-crossings of D of sign of them. That is, sw(D) = v ∈scr(D) sgn(v). Show that sw(D) does not depend on the orientation of D. Exercise 5.8.7 Show that the function fˆD (A) = (−A3 )−sw(D) is an ambient isotopy invariant of an unoriented diagram D. We can interpret fˆD (A) as a version of the Jones polynomial for unoriented links. Find the connection with the original Jones polynomial. - is an oriented diagram obtained by choosing any fixed orientation on an where D unoriented diagram D.

Lecture 6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

This lecture introduces the HOMFLYPT and the two-variable Kauffman polynomial invariants of links. We include a discussion about how the HOMFLYPT polynomial can be used to identify periodic links. The HOMFLYPT polynomial has its generalization in Conway algebras which satisfy an intriguing entropic property and lead to link invariants. We describe the Dubrovnik polynomial and observe that it is equivalent to the Kauffman polynomial (the equivalence which does not always hold for skein modules based on Kauffman and Dubrovnik polynomials). Further, we discuss the Chebyshev polynomials which have several important applications in this book. We also discuss another family of polynomials, in particular the Laguerre and Dickson polynomials, and speculate on their use in knot theory. Throughout the lecture, the authors provide a rich historical context for the development of these topics.

6.1 The Alexander Polynomial While considering quick methods of computing the Alexander polynomial1 (a classical invariant of links, compare Sect. 2.6 of Lecture 2), John H. Conway suggested a normalized form of the polynomial [Con1]. This normalized form of the Alexander polynomial is denoted by .Δ L (z), and it is now called the Conway or the Alexander-Conway polynomial. Conway showed that it satisfies the following conditions:

1 One can find this statement ironic because the classical Alexander method, which uses a certain determinant (see Lecture 2), to compute the Alexander polynomial has polynomial time complexity, while the method developed by John H. Conway has exponential time complexity. Note that computing the Jones and HOMFLYPT polynomials is NP hard [JVW]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_6

115

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6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

(i) (Initial condition) If .O is the trivial knot, then .Δ O (z) = 1. (ii) (Conway’s skein relation) .Δ L+ (z) − Δ L− (z) = zΔ L0 , where . L+, L− and . L0 are diagrams of oriented links which are identical except for the part presented in Fig. 6.1.

L+

L -

L0

Fig. 6.1: Conway’s skein triple

Conditions .(i) and .(ii) uniquely define the Conway polynomial, .Δ L (z); see [Con1, Kau1, Gi, BaMe]. James W. Alexander used the variable t in his polynomial. If √ we substitute . z = t − √1t , we obtain the normalized version of the Alexander polynomial. The skein relation now has the form: ) ( √ 1 ' .(ii ) Δ L+ − Δ L− = t − √ Δ L0 . t In fact, Alexander noted the unnormalized version of formula .(ii ') in his original 1928 paper where he introduced the polynomial [Ale1]. The Alexander polynomial was defined up to invertible elements, .±t i , in the ring of Laurent polynomials, .Z[t ±1 ], so the formula in (ii’) was not easy to use for computing the polynomial.

6.2 The HOMFLYPT Polynomial In May of 1984, V. F. R. Jones, [Jon2, Jon3, Jon4, Jon5], showed that there exists √ an invariant, .VL , of links which is a Laurent polynomial with respect to the variable . t and satisfies the following conditions: (i) (Initial condition) .VT1 (t) = 1, where .T1 is the trivial knot. ) ( √ 1 1 (ii) (Jones’ skein relation) . VL+ (t) − tVL− (t) = t − √ VL0 (t). t t The Alexander and Jones polynomials inspired further investigation of Laurent polynomials as potential invariants. This led to the discovery of a new invariant (of

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117

ambient isotopy) of oriented links. This invariant is a Laurent polynomial, .PL (x, y, z), of three variables,2 which satisfies the following conditions: (i) (Initial condition) .PT1 (x, y, z) = 1, where .T1 is the trivial knot. (ii) (HOMFLYPT skein relation) . xPL+ (x, y, z) + yPL− (x, y, z) = zPL0 (x, y, z). This invariant was discovered a few months after the Jones polynomial, in July– September of 1984, by four groups of mathematicians: W. B. R. Lickorish and Kenneth Millett, Jim Hoste, Adrian Ocneanu, and Peter Freyd and David Yetter [FHLMOY]. Independently, it was discovered in November–December of 1984 by the first author and P. Traczyk [PT1]). We call this polynomial the HOMFLYPT polynomial.3 Exercise 6.2.1 Compute the HOMFLYPT polynomial .PL (x, y, z) ∈ Z[x ±1, y ±1, z±1 ] for the trivial link of n components, .Tn . n−1 . Notice Hint: Use the skein triple from Fig. 6.2 to show that .PTn (x, y, z) = ( x+y z ) that we do not have to assume that z is invertible. It is enough to consider the subring of .Z[x ±1, y ±1, z±1 ] generated by . x ±1, y ±1, z, and . x+y z .

n-1 times

n-1 times

n-1 times

Fig. 6.2: Skein triple for trivial links

2 Most of the HOMFLYPT mathematicians started from a polynomial of three variables y, z) ∈ Z[x ±1, y ±1, z] and then assumed, with only partial justification, that z can be taken to be invertible, e.g., .z = 1 and . PL (x, y) ∈ Z[x ±1, y ±1 ]. 3 HOMFLYPT (or, as we often write: Homflypt) is the acronym after the initials of the inventors: Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter, Przytycki, and Traczyk. We also note some other names that are used for this invariant: Jones-Conway, FLYPMOTH, HOMFLY, the generalized Jones polynomial, two-variable Jones polynomial, twisted Alexander polynomial, and the skein polynomial.

. PL (x,

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6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

Remark 6.2.2 Since the HOMFLYPT polynomial was introduced and studied by several groups of mathematicians independently, various forms of the skein relation are used. Depending on the application, one form may be more convenient than the others. For example, Lickorish and Millett [LiMi] used .PL (l, m) ∈ Z[l ±1, m±1 ] with the skein relation: −1 .lPL+ (l, m) + l PL− (l, m) = mPL0 (l, m), and Hugh Morton [Mor2] used .PL (v, z) ∈ Z[v ±1, z ±1 ] with the skein relation: .

v −1 PL+ (v, z) − vPL− (v, z) = zPL0 (v, z).

Another convenient form, used in [Prz2], is .PL (a, z) ∈ Z[a±1, z ±1 ] with the skein relation: .

a−1 PL+ (a, z) + aPL− (a, z) = zPL0 (a, z) or succinctly a−1 PL+ + aPL− = zPL0 .

To go from Morton’s notation to this notation, we extend the ring .Z to .Z[i] and use the isomorphism . f : Z[i][a±1, z ±1 ] → Z[i][v ±1, z ±1 ] given by . f (a) = iv, . f (z) = −iz with the inverse map . f −1 (v) = −ia, . f −1 (z) = iz. Another notation, used in [GZ2, GZ1], is to consider framed oriented links and the ring of polynomials .Z[s±1, v ±1, x ±1, (s − s−1 )−1 ], with .s − s−1 invertible, the skein relation . x −1 PL+ − xPL− = (s − s−1 )PL0 , and the framing relation . L (1) = xv −1 L. The initial condition is as before .P O = 1. If we put . x = v and . z = s − s−1 , we get exactly Morton’s form of the HOMFLYPT skein relation of unframed oriented links. In the opposite direction, let .PL (v, z) be the HOMFLYPT polynomial in Morton’s form. We can now think that . L = L (0) is the zero framed link and define the polynomial for the framed link L with framing .w(L) to be .(xw −1 )w(L) )PL (v, z) where . z = s − s−1 . The HOMFLYPT polynomial can also be defined by starting from regular isotopy (or framed links) as described by Kauffman (Theorem 6.5.3). Example 6.2.3 We compute the polynomial .PL (a, z) of the positive Hopf link and a positive trefoil knot. First, we rewrite our skein relation to be of the form: .

PL+ = azPL0 − a2 PL− ,

and consider the diagram of the positive Hopf link, .H+ , as in Fig. 6.3a. Then: .

PH+ = azPT1 − a2 PT2 = az − a2

a + a−1 = az − (a3 + a)z −1 z

6.2 The HOMFLYPT Polynomial

119

For the diagram of the positive (right-handed) trefoil knot .3¯ 1 pictured in Fig. 6.3b,4 we obtain: .

P3¯ 1 = azPH+ − a2 PT1 = a2 (z2 − 1)PT1 − a3 zPT2 = a2 z 2 − 2a2 − a4 .

(a)

(b)

Fig. 6.3: The positive Hopf link, .H+ , and the positive trefoil knot .3¯ 1

Exercise 6.2.4 (Mirror Image Formula) Let . L¯ be the mirror image of a link L. Show that: −1 . PL ¯ (a, z) = PL (a , z). The property formulated in Exercise 6.2.4 allows us to detect the lack of amphicheirality (mirror symmetry) of a link. For example, for the negative (left-handed) trefoil knot, .31 , one has: .P31 (a, z) = a−2 z 2 − 2a−2 − a−4 , so the trefoil knot is not amphicheiral. For the figure-eight knot, .41 , which is amphicheiral, one gets 2 2 −2 . P41 = z − (a + 1 + a ). The first non-amphicheiral knot not detected by the HOMFLYPT polynomial is the .942 knot (see Fig. 6.4). The two-variable Kauffman polynomial (see Sect. 6.5) also does not detect non-amphicheirality of .942 . However, one can use the HOMFLYPT polynomial of the 2-cable of .942 to see that it is non-amphicheiral.

Fig. 6.4: The knot .942

4 The left-handed trefoil knot as shown in Rolfsen’s knot table (see Appendix C and [Rol]) is denoted by .31 ; thus, its mirror image, the right-handed trefoil knot, is denoted by .3¯ 1 .

120

6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

We will generalize the above calculations and show how to compute the change of the HOMFLYPT polynomial under the .tn -move (Fig. 6.5) and, in particular, to compute the polynomial for the torus link .T (2, n) (.T (2, 2) is the positive Hopf link and .T (2, 3) is the positive trefoil knot). −move

···

−−−−−−→

¯2 −move

···

−−−−−−−→

-half twists

2 -half twists

Fig. 6.5: Oriented .tn -move and .t¯2k -move after [Prz2] Let . Ln denote a link diagram with n positive half-twists as in Fig. 6.5 (we say that . Ln was obtained from . L0 by a .tn move). Our skein relation gives the following recursive relation: .

PLn (a, z) = azPLn−1 (a, z) − a2 PLn−2 (a, z).

This relation allows us to compute .PLn in terms of .PL1 and .PL0 . Theorem 6.2.5 ([Prz2]) .

PLn (a, z) = a n−1 Sn−1 (z)PL1 (a, z) − a n Sn−2 (z)PL0 (a, z),

where .Sn (z) are polynomials in the variable z with initial condition .S−1 = 0, .S0 (z) = 1, .S1 (z) = z, and the recursive relation .Sn (z) = zSn−1 (z) − Sn−2 (z). If one puts p n+1 −p −n−1 −1 . . z = p + p , then . Sn (z) = p−p −1 .

When the first author was writing the paper [Prz2] in 1986, he did not realize that Sn (z) are Chebyshev polynomials of the second kind (see Sect. 6.6).

Proof We proceed by induction on n. For .n = 1, 2, the formula is immediate. Assume the formula holds for .n − 1 and .n − 2. Then we have: .

PLn (a, z) = azPLn−1 (a, z) − a2 PLn−2 (a, z) ) ( = az a n−2 Sn−2 (z)PL1 (a, z) − a n−1 Sn−3 (z)PL0 (a, z) ( ) − a2 a n−3 Sn−3 (z)PL1 (a, z) − a n−2 Sn−4 (z)PL0 (a, z) ( ( ) ) = a n−1 zSn−2 (z) − Sn−3 (z) PL1 (a, z) − a n zSn−3 (z) − Sn−4 (z) = a n−1 Sn−1 (z)PL1 (a, z) − a n Sn−2 (z)PL0 (a, z), as desired.

Corollary 6.2.6 The HOMFLYPT polynomial of the torus link of type .T (2, n) is given by the following formula: .

PT (2,n) (a, z) = a n−1 Sn−1 (z) − a n Sn−2 (z)

a + a−1 . z

6.2 The HOMFLYPT Polynomial

121

Exercise 6.2.7 Compute HOMFLYPT polynomials of twist knots. Exercise 6.2.8 Consider the torus link of two components of type .T (2, 2k). Change the orientation of one component to get the link .T a (2, 2k). Find .PT a (2,2k) (a, z). Hint: Notice the similarity of computation to that of the Kauffman bracket polynomial of unoriented torus knots .T (2, n). Start from finding the change of the HOMFLYPT polynomial under .t¯2k -move that is the variant of the .t2k move with antiparallel orientation of strings; see Fig. 6.5). The calculation gives:

.

.

For example, for . k = 2 we get:

.

One can find in [Prz2] more on .tk − and .t¯2k -moves. It is often useful not to assume that the variable z is invertible. Let .R be a −1 subring5 of the ring .Z[a±1, z ±1 ] generated by .a±1 , z, and . a+a z . Let us note that z is not invertible in .R. Lemma 6.2.9 For any link L, its HOMFLYPT polynomial .PL (a, z) is in the ring .R. −1

n−1 ∈ R. Proof For the trivial link .Tn with n components, we have .PTn (a, z) = ( a+a z ) Further, if .PL+ (a, z) (respectively .PL− (a, z)) and .PL0 (a, z) are in .R, then .PL− (a, z) (respectively .PL+ (a, z)) is in .R as well. This observation enables a standard induction o to conclude Lemma 6.2.9.

One of the nice elementary properties of the HOMFLYPT polynomial is given in the following exercise. Exercise 6.2.10 (1) Show that for a knot K, the polynomial .PK (a, z) is an element of .Z[a±1, z] and .PK (a, z) − 1 is a multiple of . z 2 − (a−1 + a)2 . For example, .P41 (a, z) − 1 = z 2 − (a−1 + a)2 = a2 (P31 − 1) = a−2 (P3¯ 1 − 1) (2) Show that for any link L, the polynomial .PL ∈ R satisfies .PL (a, z) − PTcom(L) which is a multiple of . z(z −

a−1 +a z ).

5 . R is an example of a Rees algebra of .Z[a±1 ] with respect to the ideal generated by . a + a−1 ; [Eis].

122

6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

Hint: First prove (2) by induction on the number of crossings (and on the distance to the descending diagram ); see Definition 5.2 of Lecture 5. For the proof (1), compare it with the similar result for the Jones polynomial in Lecture 5. Exercise 6.2.10 can be used as a base for a more sophisticated observation related to Vassiliev-Gusarov invariants [CDM, Prz7]. Exercise 6.2.11 For a knot, K, define .P˜L (a, z) = .

-L− (a, z) ≡ lk(K0 ) -L+ (a, z) − P P

PK (a,z) . z 2 −(a−2 +a2 )

Then:

mod (a2 + 1, z),

where .lk(K0 ) is the linking number of the two-component link . L0 . Formulate a similar result for links.

6.3 Criteria for Periodic Links from the HOMFLYPT Polynomial In this section, we examine finite group actions on .S 3 which send a given link onto itself. For example, a torus link of type .(p, q) (we denote it .T (p, q)) is preserved by an action of a group .Z p ⊕ Zq on .S 3 (c.f. Fig. 6.6 and Exercise 6.3.1). Notice that if p and q are coprime, then .T (p, q) is a knot and .Z p ⊕ Zq = Z pq .

Fig. 6.6: The torus link of type .(6, 3), .T(6, 3) Subsequently, we will focus on the action of the cyclic group .Zn . We will mainly consider the case of the action on .S 3 with a circle of fixed points. The new link invariants, which we have discussed in previous sections, provide efficient criteria for examining such actions. The contents of this subsection are mostly based on papers of Kunio Murasugi [Mura4], Traczyk [Tra1] and of the first author [Prz3, Prz7, Prz17]. Exercise 6.3.1 Let .S 3 = {z1, z2 ∈ C × C : |z1 | 2 + |z2 | 2 = 1}. Let us consider an action of .Z p ⊕ Zq on .S 3 which is generated by .Tp and .Tq , where .Tp (z1, z2 ) = (e2πi/p z1, z2 ) and .Tq (z1, z2 ) = (z1, e2πi/q z2 ). Show that this action preserves torus

6.3 Criteria for Periodic Links from the HOMFLYPT Polynomial

123

links of type .(p, q). This link can be described as the following set: .{(z1, z2 ) ∈ t k S 3 | z1 = e2πi( p + p ), z2 = e2πit/q }, where t is an arbitrary real number and k is an arbitrary integer. Show that if p is coprime to q, then .Tp,q is a knot and can be parametrized by: R e t |→ (e2πit/p, e2πit/q ) ∈ S 3 ⊂ C2 .

.

Definition 6.3.2 A link is called n-periodic if there exists an action of .Zn on .S 3 which preserves the link and the set of fixed points of the action is a circle disjoint from the link. If, moreover, the link is oriented, then we assume that the generator of .Zn preserves the orientation of the link or changes it globally (i.e., on every component). New polynomial invariants of links provide strong periodicity criteria. Theorem 6.3.3 ([Prz3]) Let L be an r-periodic oriented link and assume that r is a prime number. Then the HOMFLYPT polynomial .PL (a, z) satisfies the relation: .

PL (a, z) ≡ PL (a−1, z) mod (r, zr ).

where .(r, zr ) is an ideal in .R generated by r and . zr . Proof The positive solution to the Smith conjecture [Smit, Thur1] guarantees that the fixed point set of the action of .Zr is an unknotted circle and the action is conjugate to an orthogonal action on .S 3 . In other words, if we identify .S 3 with .R3 ∪ ∞, then the fixed point set can be assumed to be equal to the “vertical” axis . z = 0 together with .∞. Then the generator of .Zr is the rotation .ϕ(z, t) = (e2πi/r · z, t), where the coordinates on .R3 come from the product of the complex plane and the real line .C × R. The r-periodic link may be represented by a . ϕ-invariant diagram (we will denote it by L), i.e., either .ϕ(L) = L or .ϕ(L) = −L depending on whether .ϕ preserves or changes the orientation of L (see Fig. 6.7).

L

Fix

Fig. 6.7: r-periodic diagram

124

6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

Now let

denote three .ϕ-invariant diagrams of

, and

links which are the same outside of the

-orbit of a fixed single crossing and which Zr

.

at the crossing differ by replacing

by

or

, respectively. Then we have the

following lemma: Lemma 6.3.4

.

o Proof (We Follow [Prz3] and [Mura4]) Let p be such a crossing in that

differs from

only at

crossings . p, ϕ(p), . . . , ϕr−1 (p); see Fig. 6.7.

Fig. 6.8: Computational tree for .PL+

We compute the contribution of leaves of a tree from Fig. 6.8 to the value of

. The contribution of the extreme left leaf

is equal to

, while the contribution of the extreme right leaf is

.

6.3 Criteria for Periodic Links from the HOMFLYPT Polynomial

125

Further, we note that all the other leaves occur r times on the tree. More precisely: the group .Zr acts on leaves of the tree pictured in Fig. 6.8 and the only fixed leaves and , while the remaining orbits have r elements (because r are o

is a prime number). −1

r Now we can proceed with the proof of Theorem 6.3.3. Since .a2r + 1 ≡ zr ar ( a+a z ) mod r, it follows that .a2r + 1 is in the ideal .(r, zr ). Hence, by Lemma 6.3.4, we get:

Theorem 6.3.5

.

o ¯ preserving the polynomial .PL The above identity allows us to compare L with . L, r modulo .(r, z ), that is: .

PL (a, z) ≡ PL¯ (a, z) mod (r, zr ).

On the other hand, we have .PL¯ (a, z) = PL (a−1, z), which enables us to conclude the o proof of Theorem 6.3.3. In order to apply Theorem 6.3.3 effectively, we need the following fact. E Lemma 6.3.6 Suppose that .w(a, z) ∈ R is written in the form .w(a, z) = vi (a)z i , i

where .vi (a) ∈ Z[a±1 ]. Then .w(a, z) ∈ (r, zr ) if and only if for any .i ≤ r the coefficient −1 r−i ). .vi (a) is in the ideal .(r, (a + a ) −1

r−i z r , it follows that .(a + a−1 )r−i z i ∈ (r, z r ). Proof Since .(a + a−1 )r−i z i = ( a+a z ) −1 r−i Therefore, if .vi (a) ∈ (r, (a + a ) ), then .w(a, z) ∈ (r, zr ). Conversely, suppose that r r .w(a, z) ∈ (r, z ), that is, .w(a, z) ≡ z w(a, z)mod r for some .w(a, z) ∈ R. The element E a+a−1 j ( z ) v j (a), .w(a, z) can be uniquely written as a sum .w(a, z) = zv(a, z) + j ≥0

where .v(a, z) ∈ Z[a±1, z] and .v j (a) ∈ Z(a±1 ). Thus, for .i ≤ r we have .vi (a) ≡ o (a + a−1 )r−i v r−i (a) mod r and finally .vi (a) ∈ (r, (a + a−1 )r−i ) for .i ≤ r. Example 6.3.7 Consider the knot .11388 , in Perko’s notation [Perk3] (see Fig. 6.9). The HOMFLYPT polynomial .P11388 (a, z) is equal to: (3 + 5a2 + 4a4 + a6 ) + (−4 − 10a2 − 5a4 )z 2 + (1 + 6a2 + a4 )z 4 − a2 z 6 .

.

One can check that for .r ≥ 7 the difference .P11388 (a, z) − P11388 (a−1, z) is not in the ideal .(r, zr ). Therefore, from Theorem 6.3.3, it follows that the knot .11388 is not r-periodic for .r ≥ 7.

126

6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

11388

Fig. 6.9: The knot .11388

In Theorem 6.3.3, we have applied an idea which Traczyk used to the Jones polynomial [Tra1]. The following corollary is a slight generalization of the original result of Traczyk. Corollary 6.3.8 If L is an r-periodic oriented link (r prime number), then its Jones polynomial satisfies the relation: −1 .VL (t) ≡ VL (t ) mod (r, t r − 1). Exercise 6.3.9 Show that if applying our method to Alexander polynomial, we obtain the following version of a theorem of Murasugi: Δ L (t) ≡ ΔrL∗ (t)(1 + t + t 2 + . . . + t λ−1 )r−1

.

mod r

where . L∗ is the quotient of an r-periodic link L and .λ is the linking number of L and z axis. Hint: Construct the binary computational tree for the Alexander polynomial of . L∗ and the associated binary tree for Alexander polynomial of L modulo r. Use the fact that the Alexander polynomial of a split link (a link with components that can be separated into disjoint balls) is equal to zero; compare Fig. 6.8. Exercise 6.3.10 If .r = 2, then the formula from the previous exercise is as follows: Δ L (t) ≡ Δ2L∗ (t)(1 + t + t 2 + . . . + t λ−1 )

.

mod 2.

Show that if L is a strongly .+ amphicheiral link, then modulo 2 the polynomial Δ L (t) is a square of another polynomial. A knot (or oriented link) in .R3 is called strongly plus amphicheiral if it has a realization (i.e., concrete realization of an

.

6.4 Conway Algebras and Entropy Condition

127

embedding) in .R3 which is preserved by a (changing orientation) central symmetry .((x, y, z) → (−x, −y, −z)); “plus” means that the involution is preserving orientation of the link. Richard Hartley and Akio Kawauchi [HaKaw] proved that .Δ L (t) is a square in general, but we do not know an elementary proof of this result. Exercise 6.3.11 Show that if a ball bouncing from the walls of a cubic room moves along a closed orbit, forming a knot, K, then the Alexander polynomial .ΔK (t) is, modulo 2, a square of another polynomial. For a solution, refer to the papers [JoPr, Prz17] where we introduce the theory of billiard knots. More information on periodicity criteria from knot polynomials can be found in [Prz9, PS2, Tra2, Tra3, Yok1, Yok2, Yok3, Yok4].

6.4 Conway Algebras and Entropy Condition Instead of looking for polynomial invariants of links related to the Conway skein triple shown in Fig. 6.1, we can approach the problem from a more general point of view. Namely, we can look for universal invariants of links which have the following property: a given value of the invariant for . L+ and . L0 determines the value of the invariant for . L− , and similarly: if we know the value of the invariant for . L− and . L0 , we can find its value for . L+ . The invariants with this property are called Conway-type invariants. We will develop these ideas in the present lecture of the book which is based mainly on a joint work of Traczyk and the first author [PT1, Prz1] (see also [NiPr]). Let us consider the following general situation involving an universal algebra, A,6 with a universe, A, and a countable number of 0-argument operations (fixed elements) denoted by .a1, a2, . . . , an, . . . and two 2-argument operations denoted by . | and . * . We would like to construct an invariant w of oriented links with values in A which satisfies the following conditions: .

.

w L+ = w L− |w L0 and w L− = w L+ * w L0 and wTn = an,

where .Tn is a trivial link of n components. 6 A universal algebra (or abstract algebra) is a set, called a universe, with a set of n-ary operations, ≥ 0.

.n

128

6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

The operation . | is meant to recover values of the invariant w for . L+ from its values for . L− and . L0 , while the operation . ∗ is supposed to recover values of w for . L− from its values for . L+ and . L0 . Definition 6.4.1 We say that .A = (A; a1, a2, . . . , |, *) is a Conway algebra if the following conditions are satisfied: } C1 an |an+1 = an . initial values properties C2 an * an+1 = an

.

) C3 (a|b)|(c|d) = (a|c)|(b|d) ⎪ ⎬ ⎪ C4 (a|b) * (c|d) = (a * c)|(b * d) transposition or entropy properties ⎪ C5 (a * b) * (c * d) = (a * c) * (b * d) ⎪ ⎭ C6 (a|b) * b = a C7 (a * b)|b = a

.

}

inversion properties.

We will now formulate the main result of this subsection. Theorem 6.4.2 ([PT1]) For a given Conway algebra, .A, there exists a uniquely determined invariant, w, of oriented links which to any link, L, (up to ambient isotopy) associates an element .w L ∈ A and satisfies the following conditions: .

.

(1) wTn = an – initial conditions

(2) w L+ = w L− |w L0 (3) w L− = w L+ * w L0

} – Conway relations

For a proof (long and technical), we refer to the first author’s original paper [PT1] or [Prz1]. Now we briefly discuss the geometric interpretation of conditions .C 1 − C7 in the definition of Conway algebra. Conditions C1 and C2 reflect relations between trivial links of n and .n + 1 components. The diagrams of the links in these relations are pictured in Fig. 6.2. Relations C3, C4, and C5 are obtained when we perform a calculation of a link invariant at two crossings of the diagram in different order. Relations C6 and C7 illustrate the fact that we need the operations . | and .* to be inverses of one another.7 7 This can be formalized as follows. Denote, after [Prz23], by .Bin(X) the set of all binary operations × X → X. .Bin(X) can be given a monoid structure as follows: let .∗1, ∗2 ∈ Bin(X), and then the composition .∗1 ∗2 is given by . a(∗1 ∗2 )b = (a ∗1 b) ∗2 b. The identity element .∗0 is defined by . a ∗0 b = a. According to this definition, . | and .∗ are inverse elements in the monoid .Bin(A), that is, . |∗ = ∗0 = ∗|. .X

6.4 Conway Algebras and Entropy Condition

129

In the definition of a Conway algebra, we have introduced seven conditions. We gave all of them for aesthetic and practical reasons (we wanted to display the symmetry between the two relations). These conditions, however, are not independent from each other: Lemma 6.4.3 In the definition of the Conway algebra, there are the following dependencies between conditions .C 1 and C7: (a) C1 and C6 ⇒ C2 (b) C2 and C7 ⇒ C1 (c) C6 and C4 ⇒ C7 (d) C7 and C4 ⇒ C6 . (e) C6 and C4 ⇒ C5 ( f ) C7 and C4 ⇒ C3 (g) C5, C6 and C7 ⇒ C4 (h) C3, C6 and C7 ⇒ C4 We will prove, as examples, the implications (a) and (c). (a) C1 ⇔ ⇒ an |an+1 = an C6

(an |an+1 ) * an+1 = an * an+1 ⇒ ⇔ an = an * an+1 C2. . (c) C6 ⇒ C4

(a|(b|a)) * (b|a) = a

⇔

(a * b)|((b|a) * a) = a (a * b)|b = a C7.

⇒ ⇔

C6

Remark 6.4.4 When the first author worked with Traczyk on the paper [PT1], they did not know that magmas satisfying the axiom C5 of Conway algebras have a long history in the theory of nonassociative algebras. Probably, the first people considering the equation .(a · b) · (c · d) = (a · c) · (b · d) were Celestin Burstin and Walter Mayer in 1929 [BuMa]. The equation was also considered by Anton Sushkevich in 1937 [Sus] probably influenced by [BuMa]. A magma satisfying the property is often called an entropic magma (the term coined in 1949 by Ivor M.H. Etherington [Eth] ). The word entropic refers to inner turning [RoSm]. Other names for the property .(a · b) · (c · d) = (a · c) · (b · d) are medial, alternation, bi-commutative, bisymmetric, commutative, surcommutative, and abelian. The Conway algebra, .AHOMFLYPT , yielding the HOMFLYPT polynomial .PL (x, y, z) n−1 , .a|b = z b− y a, and .a∗b = z b− x a. is given by: . A = Z[x ±1, y ±1, z ±1 ], .an = ( x+y z ) x x y y Note that:

130

6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

( (a ∗ b)∗(c ∗ d) =

.

) ( ) z z x x x2 z2 xz b − a ∗ d − c = 2 a + 2 d − 2 (b+ c) = (a ∗ c)∗(b∗ d), y y y y y y y

as expected. The following classical result from entropic magmas can be used in knot theory. Recall that a magma .(X, ∗) is called a quasigroup if for any triple of ordered elements .a, b, c each pair uniquely defines the third, satisfying .a ∗ b = c. Theorem 6.4.5 ([Murd, Toy]) If .(X, ∗) is an entropic quasigroup, then X has an abelian group structure .(X, +) such that .a ∗ b = f (a) + g(b) + c where .a, b, c ∈ X and [ . f , g : X] → X are commuting group automorphisms. Equivalently, .(X, ∗) is a ±1 ±1 -module. .Z x , xg f In particular, the result of Adam Sikora [Sik1] about invariants coming from quasigroup Conway algebras can be interpreted as a generalization of the MurdochToyoda theorem.

6.5 The Two-Variable Kauffman Polynomial It is a natural question to ask whether the three diagrams, . L+ , . L− , and . L0 , which have been used to build Conway-type invariants can be replaced by other diagrams. In fact, at the turn of December and January of 1984/1985, Krzysztof Nowiński (of Warsaw University) suggested considering another diagram apart from . L+ , . L− , and . L0 , namely, the diagram obtained by smoothening . L+ without preserving the orientation of . L+ (Fig. 6.10). At that time, however, the first author and Traczyk did not make any effort to exploit this idea to get new invariants of links; it is likely they were discouraged by a lack of a natural orientation on . L∞ .

+

Fig. 6.10: A smoothing of a crossing that is not agreeing with the original orientation In the early spring of 1985, Robert D. Brandt, W. B. R. Lickorish, K. C. Millett, [BLM] and, independently, Chi Fai Ho [Ho] proved that four diagrams of unoriented links pictured in Fig. 6.11 (the .+ sign in . L+ does not denote the sign of the crossing, even if the sign is defined) can be used to construct invariants of unoriented links.

6.5 The Two-Variable Kauffman Polynomial

+

−

131

0

∞

Fig. 6.11: Skein quadruple

Theorem 6.5.1 There exists a uniquely determined invariant Q which to any (ambient) isotopy class of invariant links associates an element of .Z[x ±1 ]. The invariant Q satisfies the following conditions: (1) If .O is the trivial knot, then .Q O (x) = 1. (2) .Q L+ (x) + Q L− (x) = x(Q L0 (x) + Q L∞ (x)), where unoriented diagrams of links . L+, L−, L0, and . L∞ are identical outside of the parts pictured in Fig. 6.11. The polynomial .Q L (x) has a number of features similar to those of the HOMFLYPT polynomial .PL (v, z). The proof of these properties is left for the reader as an exercise. Exercise 6.5.2 Prove that: ( ) n−1 , where .Tn is the trivial link of n components. (a) .QTn (x) = 2−x x (b) .Q L1 #L2 (x) = Q L1 (x)Q L2 (x), where .# denotes the connected sum of links; see Lecture 5, Exercise 5.2.4. ( ) n−1 (c) .Q L1 uL2 = μQ L1 (x)Q L2 (x), where . μ = QT2 (x) = 2−x . x (d) .Q L (x) = Q L¯ (x), where . L¯ is the mirror image of the link L. (e) .Q L (x) = Q m(L) (x), where .m(L) is the mutant of the link L (see Lecture 5). (f) .Q L (x)(−2) = (−2)com(L)−1 . Now we will discuss Kauffman’s approach [Kau2, Kau6] which, in particular, allows to generalize the polynomial .Q L (x) to an invariant which often distinguishes mirror images. This approach is based on Kauffman’s idea that instead of considering diagrams modulo equivalence (i.e., diagrams up to all three Reidemeister moves, corresponding to ambient isotopy of links), one can consider diagrams up to the equivalence relation which is based on the second and third Reidemeister moves (i.e., regular isotopy). This way one does not get an invariant of links but often a simple correction/balancing allows to construct an ambient isotopy invariant (as was done in Lecture 5 where the Jones polynomial was constructed from the Kauffman bracket).8 For historical record, we should add that the Kauffman polynomial of two variables was introduced before the Kauffman bracket (see Lecture 2). 8 Kauffman’s approach has a good interpretation in terms of framed links (the approach is used in Lecture 11 on skein modules of three-dimensional manifolds).

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6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

The idea of Kauffman is based on the observation that the trivial knot can be represented (up to ambient isotopy) by different regular isotopy classes of diagrams, and to any such a class, we can associate different values of some invariant. To any diagram .T1 representing the trivial knot, Kauffman associates the monomial .aw(T1 ) . Subsequently, the Kauffman definition of invariants is similar to the definition of the Conway polynomial, .Δ L (z), and the Brandt-Lickorish-Millett and Ho polynomial, .Q L (x). Now we show how, using Kauffman’s approach, the Conway polynomial can be generalized to the HOMFLYPT polynomial and the polynomial .Q L can be modified to another invariant which is called the Kauffman polynomial of two variables (or shortly the Kauffman polynomial). Theorem 6.5.3 ([Kau2]) 1. There exists a uniquely defined invariant of regular isotopy of oriented diagrams, denoted by .(RL (a, z)), which is a polynomial in .Z[a±1, z ±1 ] and which satisfies the following conditions: a. . RT1 (a, z) = aw(Ta ) , where .T1 is a diagram of a knot isotopic to the trivial knot. b. . RL+ (a, z) − RL− (a, z) = zRL0 (a, z). 2. For any diagram L, we consider a polynomial .G L (a, z) = a−w(L) RL (a, z). Then .G L (a, z) is an invariant of ambient isotopy of oriented links, and it is equivalent to the HOMFLYPT polynomial, that is: −1 a . G L (a, z) = PL (x, y) where x = , y = z za

.

In a similar way, Kauffman generalizes the Brandt-Lickorish-Millett and Ho polynomial, this time obtaining a totally new polynomial, the Kauffman polynomial.

Theorem 6.5.4 ([Kau6]) (1) There exists a uniquely defined invariant .Λ which to any regular isotopy class attaches a polynomial in .Z[a±1, z ±1 ] and which satisfies the following conditions: (a) .ΛT1 (a, z) = aw(T1 ) , (b) .Λ L+ + Λ L− = z(Λ L0 + Λ L∞ ). (2) For any oriented diagram D, we define .FD (a, z) = a−w(D) ΛD (a, z). Then . FD (a, z) is an invariant of (ambient) isotopy classes of oriented links, and it is a generalization of the polynomial Q, that is, .Q L (x) = FL (1, x). Now we check some elementary properties of the Kauffman polynomial.

6.5 The Two-Variable Kauffman Polynomial

133

Exercise 6.5.5 Show that: −1 −z

(a) .FTn = ( a+az

)n−1 ,

(b) .FL1 #L2 (a, z) = FL1 (a, z) · FL2 (a, z). (c) .FL1 uL2 (a, z) = μFL1 (a, z) · FL2 (a, z) where . μ = FT2 = the invariant on the trivial link with two components.

a−1 +a z

− 1 is the value of

(d) .FL (a, z) = F−L (a, z). (e) .FL (a, z) = FL (a−1, z). (f) .FL (a, z) = Fm(L) (a, z) where .m(L) is a mutant of the link L. The polynomial .Λ does not depend on the orientation of the diagram D. Therefore, the dependence of the Kauffman polynomial F on the orientation of components of D is easy to describe—this is because .FD (a, z) differs from .ΛD (a, z) by a power of a only. Lemma 6.5.6 Let . D = {D1, D2, . . . , Di, . . . , Dn } be a diagram of an oriented link with n components and let . D ' := {D1, D2, . . . , −Di, . . . , Dn }. We set .λi = 1E sgn(p j ) where the summation is over all crossings of . Di with . D − Di . The num2 ber .λi is called the linking number of . Di and . D − Di and it is often denoted by th .lk(Di , D − Di ); it is an ambient isotopy invariant of D (where . Di is the .i component in the chosen ordering of components of D); this follows easily from considering Reidemeister moves. Then: '

.

FD' (a, z) = a−w(D ) aw(D) FD (a, z) = a4λi FD (a, z).

Let us note that the above lemma implies that the Kauffman polynomial gives only as much information about orientations of components of D as comes from the linking numbers of its components . Di with complements . D − Di . The Kauffman polynomial is very useful for distinguishing a given link from its mirror image (achirality of a link). However, there are examples, such as the knots .942 and .1071 (according to Rolfsen’s notation [Rol]; see Fig. 6.12) which are not

9 42

10 71

Fig. 6.12: Knots with symmetric Kauffman polynomials

134

6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

isotopic to their mirror images, but still have the same Kauffman polynomials as their mirror images. The Kauffman polynomial is also a generalization of the Jones polynomial. Exercise 6.5.7 ([Lic4]) Prove that Eq. 6.1 holds: VL (t) = FL (t 3/4, −(t −1/4 + t 1/4 )).

.

(6.1)

Hint: Compare the polynomial .ΛD to the Kauffman bracket . . Follow the reasoning in which we got the Jones polynomial from the Kauffman bracket, getting first: −1 . + = (A + A )( + ). Exercise 6.5.8 Find the Kauffman polynomial .Λ L (a, z) for the Hopf link .H+ , the positive trefoil knot .3¯ 1 , and the figure-eight knot, .41 . Exercise 6.5.9 Show the relation between the Fox 3-colorings and the Kauffman polynomial, that is, .col3 (L) = 3|FL (1, −1)|. Let us finish this section by observing (in analogy to Theorem 5.4.4 that the Brandt-Lickorish-Millett-Ho polynomial (so the Kauffman polynomial) is also stronger than the Fox 5-coloring invariant. We have: Exercise 6.5.10 Show that .col5 (L) = 5|Q((e2πi/5 + e−2πi/5 )| 2 . Hint: Interpret both sides of the equation in the exercise using the first homology of the of .S 3 branched along L. We would also challenge a reader to find totally elementary proof of the formula in Exercise 6.5.10.

6.5.1 The Dubrovnik Polynomial In the late summer of 1985, Kauffman constructed yet another skein relation using link diagrams . L+ , . L− , . L0 , and . L∞ as in Fig. 6.11. The relation is very similar to the original Kauffman polynomial relation. Kauffman described the polynomial on a postcard to Lickorish sent from Dubrovnik in September 1985. He expected, initially, that this was a new polynomial invariant of links, independent from F.9 Let us denote the new polynomial by .Λ∗ in an unoriented version and .F ∗ in an oriented version. Here is the definition:

9 I think I was at Oberwolfach conference when Lickorish got the postcard; my memory is fading now so maybe some reader has clearer recollection.

6.5 The Two-Variable Kauffman Polynomial

135

Definition 6.5.11 (1) The polynomial .Λ∗D (a, z) ∈ Z[a±1, z ±1 ], for unoriented diagrams, is defined by the initial condition .ΛT∗1 = 1 and the skein relation: Λ∗D+ − Λ∗D− = z(Λ∗D0 − Λ∗D∞ ).

.

→ − − ∗ −w(D) Λ∗ (a, z), where → (2) .F→ . D is the diagram D equipped with arbitrary − (a, z) = a D D → − → − (but fixed) orientation and .w( D) is the write of . D) (see Lecture 5, Definition 5.4.2).

Substituting (2) to (1), in the form .Λ∗D (a, z) = aw(→D) F -∗ (a, z), we get: D

Proposition 6.5.12 The polynomial .F ∗ satisfies a recursive condition: → −

.

→ −

→ −

∗ w(D − ) ∗ w(D 0 ) ∗ ∗ aw(D+ ) F→ F→ FD0 − aw(D ∞ ) F→ − −a − = z(a − ). D+

D−

D∞

or equivalently: .

aF -∗ − a−1 F -∗ = z(F -∗ − aw(D∞ )−w(D0 ) F -∗ ). -

D+

D−

-

D0

D∞

Recall that the Kauffman polynomial for oriented links satisfies an analogous relation: .

aFD- + + a−1 FD- − = z(FD- 0 + aw(D∞ )−w(D0 ) FD- ∞ ). -

-

To convert the Kauffman polynomial to Dubrovnik (change .+ to .−), the standard way is to try to substitute ia for a and .−iz for z. To understand the coefficient of .FD- ∞ , - ∞ ) − w( D - 0 ). we should understand better .w( D - and .sgn(v) its sign. Proposition 6.5.13 Let v be a crossing of an oriented diagram . D - ∞, To orient . D∞ , we have to make some choices, and to make our convention for . D we consider two cases depending on whether v involves two components of D (mixed crossing) or only one component (self-crossing). - i and . D - j of . D. - We (1) Assume that v is a mixed crossing between components . D define an orientation on . D∞ by first changing the orientation of . D j and then smoothing v according to this new orientation. We have: .

- 0 ) = w( D) - − sgn(v) and w( D - ∞ ) = w( D) - − 4lk( D - j, D - −D - j ) + sgn(v). w( D

Thus: .

- ∞ ) − w( D - 0 ) = −4lk( D - j, D - −D - j ) + 2sgn(v). w( D

136

6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

In particular: .

- ∞ ) ≡ w( D - 0) + 2 w( D

mod 4.

- ∞ ) = com( D - 0 ) = com( D) - − 1. Furthermore, the number of components .com( D - i of . D. - Then . D - i splits at (2) Assume that v is a self-crossing of a component . D - 0 , say . D - a and . D - b . To get orientation on v into two oriented components in . D → − ← − . D∞ , we change the orientation of . D b to . D b . Since . D∞ has the same number → − of components as D, then we can choose an orientation on . D∞ (. D ∞ ) so that it → − ← − agrees with the orientations of . D a and the orientation of . D b . We have: .

- 0 ) = w( D) - − sgn(v) and w( D - ∞ ) = w( D) - − 4lk( D - b, D -0 − D - b ) − sgn(v). w( D

In particular: .

- ∞ ) ≡ w( D - 0 ) mod 4. w( D

- ∞ ) = com( D - 0 ) − 1 = com( D). Furthermore, the number of components .com( D Corollary 6.5.14 In both cases of Proposition 6.5.12, we have: .

- ∞ ) − w( D - 0 ) ≡ 2(com( D - ∞ ) − com( D - 0 )) + 2 w( D

mod 4.

We are ready to formulate the main result connecting Kauffman and Dubrovnik polynomials. Theorem 6.5.15 ([Lic8]) (1) Two oriented links have the same Dubrovnik polynomial if and only if they have the same Kauffman polynomial. (2) If we extend the ring .Z[a±1, z ±1 ] to . Z[i][a±1, z ±1 ], then .

( ) ¯ ∗ com(D)−1 − (ia, −iz). F→ F→ − (a, z) = (−1) D D

Proof The formula of (2) follows from Proposition 6.5.12 and Corollary 6.5.14. The initial condition .FT1 = 1 = FT∗1 holds because .com(T1 ) − 1 = 0. Part (1) follows from Part (2). o Exercise 6.5.16 Check the theorem for trivial links. Remark 6.5.17 The Kauffman and Dubrovnik polynomials generalize to skein modules of three-dimensional manifolds. However, these skein modules are not always the same, for example, they differ for the real projective space .RP3 ; see Lecture 11 and [Mro4].

6.6 Chebyshev Polynomials

137

6.6 Chebyshev Polynomials Chebyshev polynomials are named after the Russian mathematician Pafnuty Lvovich Chebyshev. The classical book on Chebyshev polynomials [MH] states that “Chebyshev polynomials are everywhere dense in numerical analysis.” We can make a similar statement about knot theory: Chebyshev polynomials are dense in modern knot theory. Soon after the Jones polynomial and its generalizations were discovered, the first author observed that they behave in an interesting manner when a link is modified by a .tk -move [Prz2]. Later, the behavior under .tk -moves was codified by Chebyshev polynomials. From that time, Chebyshev polynomials have been frequently used in knot theory. Among the vast amounts of research in knot theory that uses Chebyshev polynomials, an important instance is seen in Jones-Wenzl idempotents and their role in Lickorish’s construction of the Witten-ReshetikhinTuraev 3-manifold invariants (see [Lic8] and Lecture 16). Furthermore, they are an important tool in the product-to-sum formula for the multiplication of curves in the Kauffman bracket skein algebra of the torus [FrGe]. In fact, it is conjectured that the Chebyshev basis (see Lecture 14) of the Kauffman bracket skein algebra of a surface is always positive [Thur1, BMPSW, Bou]. They also appear in a presentation of handle sliding relations in the Kauffman bracket skein module of 3-manifolds [BLP]. In this account of Chebyshev polynomials, we mostly follow [MH, Lic8]. We should stress that different conventions are used in classical books (e.g., [MH]) and in knot theory [Lic8]. We start from the convention popularized by Lickorish: Definition 6.6.1 We define the Chebyshev polynomial of the first kind, .Tn (d) by the initial conditions .T0 (d) = 2, .T1 (d) = d, and the recursive relation .Tn (d) = dTn−1 (d)−Tn−2 (d). T in Chebyshev polynomial refers to the old spelling Tchebycheff. Using the same recursive relation, written in the form .dTn (d) = Tn−1 (d) + Tn+1 (d), we can extend the Chebyshev polynomial .Tn (d) to negative indexes and observe that .T−n (d) = Tn (d). If .d = 2 cos(θ), then .Tn (2 cos(θ)) = 2 cos(nθ). In the Chebyshev polynomial considered in [MH], the nth term .Tn' (x) is equal to .2Tn (d), For .d = 2x. Thus, the initial conditions are .T0'(x) = 1, .T1'(x) = x, and the recursive relation is given by ' ' ' ' .Tn (x) = 2xT n−1 (x) − Tn−2 (x). Finally, in [MH] notation, .Tn (cos θ) = cos(nθ). Notice the trigonometric identity: .cos(nθ) + cos((n − 2)θ) = 2 cos(θ) cos((n − 1)θ). Definition 6.6.2 We define the Chebyshev polynomial of the second kind .Sn (d) by the initial conditions .S0 (d) = 1, .S1 (d) = d, and the recursive relation .Sn (d) = dSn−1 (d) − Sn−2 (d). S in the notation of this Chebyshev polynomial refers to sinus function to which it is related (see below). Often the notation .Un (d) is used for the Chebyshev polynomial of the second kind. . .

Sn (d) can also be extended to negative indexes, and we get .S−n (d) = −Sn−1 (d) with S−1 = 0.

138

6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

If .d = 2 cos(θ), then .Sn (2 cos(θ)) =

sin((n+1)θ) sin(θ) .

For the Chebyshev polynomial of the second kind, considered in [MH], the nth term .Sn' (x) is equal to .Sn (d) for .d = 2x. Thus, we have the initial conditions .S0' (x) = 1, ' (x) − S ' (x). Finally, and .S1' (x) = 2x, and the recursive relation .Sn' (x) = 2xSn−1 n−2 sin((n+1)θ) ' . Notice the trigonometric identity: .sin((n + 1)θ) + sin((n − . Sn (cos(θ)) = sin(θ) 1)θ) = 2 cos(θ) sin(nθ). To deal effectively with the Chebyshev polynomials .Tn (d) and .Sn (d), we consider the substitution, motivated by the trigonometric substitutions, .d = p + p−1 . We get p n+1 −p −n−1 and .Tn (d) = pn + p−n . This leads immediately to the formulas . Sn (d) = p−p −1 .Tn (d) = Sn (d) − Sn−2 (d) and

.

Sn (D) =

[ n2 ] E i=0

Tn−2i − e =

n−1 [E 2 ]

Tn−2i + e,

i=0

where .e = 0 for n odd and 1 for n even. We should comment here that the set {Tn (x)}n≥0 does not form a basis of .Z[x] because .T0 (x) = 2 and the free part is always divisible by 2. To have a basis, we need the initial condition to be equal to 110 . We also have the following useful formula:

.

Tn2 (d) − d 2 = Sn (d)Sn−2 (d)(d 2 − 4)

.

which follows easily by considering .d = p + p−1 : 2 2 n −n )2 − (p + p−1 )2 = p2n + p−2n − p2 − p−2 = .Tn (d) − d = (p + p n+1 −n−1 n−1 .(p −p )(p − p−n+1 ) = Sn (d)(p − p−1 )Sn−2 (d)(p − p−1 ) = 2 . Sn (d)Sn−2 (d)(d − 4). Remark 6.6.3 Notice that . pn Sn (d) = 1 + p2 + . . . + p2n = [n + 1] p2 ; thus, Chebyshev polynomials will play an important role in the study of the noncommutative plane (or noncommutative torus, which is related to the Kauffman bracket skein algebra of the thickened torus, .T 2 × [0, 1]; see papers [BuPr, FrGe]). Exercise 6.6.4 Let .Pm (x) be a polynomial (over any commutative ring with identity) satisfying Chebyshev recursive relation: .Pm (x) = xPm−1 (x) − Pm−2 (x) and arbitrary initial data, .P0 (x) = y, .P1 (x) = z. Show that: .

Pm (x) = zSm−1 (x) − ySm−2 (x),

where .Sn is the Chebyshev polynomial of the second kind. 10 The bracelet .Tˆn (d) is the modification of .Tn (d) so that .Tˆ0 (d) = 1 and .Tˆn (d) = Tn (d) for .n > 0. The reason for such a modification is that .2, T1 (d), T2 (d), . . . is not a basis of the .Z-module .Z[d], while .1, T1 (d), T2 (d), . . . is a basis.

6.6 Chebyshev Polynomials

139

For Lecture 14 the most important identity is the product-to-sum identity: Tm (d)Tn (d) = Tm+n (d) + T |m−n | (d).

.

And for the positivity of S-Chebyshev decoration, we start from: .

Sm (d)Sn (d) = S |n−m | (d) + S |n−m |+2 (d) + . . . + Sn+m−2 (d) + Sn+m (d) n+m−|n−m| 2

=

E

S |n−m |+2i

i=0

=

E

Si (d),

i=Im, n

where . Im,n = {i | |n − m| ≤ i ≤ m + n, m + n − i is even}.

6.6.1 Chebyshev Polynomials as Orthogonal Polynomials In knot theory, it seems that we do not use the fact that for a properly defined inner product, Chebyshev polynomials are orthogonal polynomials. With this we should use the original Chebyshev polynomial of the first kind .Tn' (x) so that .T0'(x) = 1, ' ' ' ' .T (x) = x, and the recursive relation is given by .Tn (x) = 2xT 1 n−1 (x) − Tn−2 (x). We have .Tn' (cos(θ)) = cos(nθ). Let us start from more general definition (we follow [Vre] in this account): Definition 6.6.5 Let .C (a, b) be the set of continuous, complex valued functions defined on the compact interval .[a, b]. Let w be a fixed continuous function on .[a, b] such that .w(x) > 0 and define an inner (Hermitian) product on .C (a, b) by: ∫ .

b

< f , g> =

f (x)g(x)w(x)dx.

a

One looks for various orthogonal functions (“bases” in a completion sense). One starts from the weight .w(x) = 1 to get Fourier coefficients. If we look for a polynomial “basis,” we have Legendre polynomials, obtained from the base .1, x, x 2, x 3, . . . using the Gram-Schmidt procedure, still with .w(x) = 1. If we choose .w(x) = e−x , we 2 are led to Laguerre polynomials. With weight .w(x) =√e−x , we get the Hermite polynomials. Finally, with the weight function .w(x) = 1/ 1 − x 2 , we get the Chebyshev polynomials—.T ' (x). The integrals giving the inner product have the form: ∫ .

< f , g> =

1

−1

f (x)g(x) √

dx 1 − x2

.

140

6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

It can be proven that the polynomials named after Laguerre, Hermite, and Chebyshev actually constitute complete orthogonal systems in their respective spaces [MH]. It is a challenge to the reader to find the use of these polynomials in knot theory.11

6.6.2 Dickson Polynomials The straight generalization of the Chebyshev polynomials are the Dickson polynomials of two variables, x and .α, introduced by Leonard Dickson in his 1896 PhD thesis [Dic]. Definition 6.6.6 For nonnegative n, we define Dickson polynomials as follows: (1) The Dickson polynomial of the first kind, . Dn (x, α), is given by the initial conditions . D0 (x, α) = 2 and . D1 (x, α) = x, and by the recurrence relation: .

Dn (x, α) = xDn−1 (x, α) − αDn−1 (x, α).

(2) The Dickson polynomial of the second kind, .En (x, α), satisfies the same recurrence relation, but differs by initial condition .E0 (x, α) = 1, .E1 (x, α) = x. Exercise 6.6.7 If .α is invertible, then Dickson polynomials are defined for positive and negative indexes. Show that . D−n (x, α) = α−n Dn (x, α) and .E−n (x, α) = −α n−1 En−2 (x, α). After substituting . x = p + αp , we get, in analogy to Chebyshev polynomials: Exercise 6.6.8 Show that: (1) . Dn (x, α) = pn + (2) .En (x, α) =

αn pn ,

p n+1 −α n+1 p −(n+1) . p−αp −1

Exercise 6.6.9 Show that . Dn (x, α) = En (x, α) − En−2 (x, α). The Dickson polynomial recurrence relation . Dn (x, α) = xDn−1 (x, α)−αDn−2 (x, α) corresponds to the skein relation of the HOMFLYPT polynomial .PL (x, y, z): .

PL+ =

z y PL0 − PL− . x x

More about the Dickson polynomials can be found in [LMT, Mul]. 11 The intriguing connection between the Laguerre polynomial and the matching polynomial from graph theory was observed by Ivan Gutman in 1979 [Gut, SDLG].

6.7 Exercises

141

6.7 Exercises Exercise 6.7.1 Compute the Alexander-Conway polynomial of the left-handed trefoil using Conway’s skein relation. Exercise 6.7.2 (i) Prove that the Alexander polynomial of the unlink of two components is equal to zero. (ii) Use the same method as in (i) to prove that the Alexander polynomial of the unlink of n-components is equal to zero. (iii) Prove that the Alexander polynomial of a split link (L1 u L2 ) is equal to zero. Exercise 6.7.3 (i) Prove the following equation:

.

(ii) Use (i) to compute the HOMFLYPT polynomial of the 72 knot as shown in Fig. 6.13.

Fig. 6.13: The 72 knot

Exercise 6.7.4 (i) Compute the two-variable Kauffman polynomial of the trefoil knot 31 . (ii) Use Exercise 6.5.5 to compute the two-variable Kauffman polynomial of the square knot 31 #3¯ 1 .

142

6 The HOMFLYPT and the Two-Variable Kauffman Polynomial

Exercise 6.7.5 Consider the Chebyshev polynomial of the second kind Sn (d) and use the substitution d = p + p−1 to demonstrate the product-to-sum formula: .

Sn (d)Sm (d) = Sn+m (d) + Sm−1 (d)Sn−1 (d).

Exercise 6.7.6 (F) Consider Fibonacci numbers, F0 = 0, F1 = 1, F2 = 1, F3 = 2, F4 = 3, F5 = 5, . . . Fn+2 = Fn+1 + Fn . Show that F2n = Sn−1 (3). Furthermore, F2n = Fn (Fn−1 + Fn+1 ). (L) Consider Lucas numbers L0 = 2, L1 = 1, L2 = 3, L3 = 4, L4 = 7, . . ., Ln+2 = Ln+1 + Ln . Show that L2n = Tn (3), where Tn is the Chebyshev polynomial of the first kind. Solution: F2n = F2n−1 + F2n−2 = 2F2n−2 + Fn−3 = 3F2n−2 − F2n−4 . Thus, F2n and Sn−1 (3) satisfy the same recursive relation. Noting that F2 = 1 = S1 (3) and F4 = 3 = S1 (3), we conclude the formula from the exercise. To prove F2n = Fn (Fn−1 + Fn+1 ), we first write Fn+k = an Fn + bn Fn−1 and find (inductively) that an = Fk+1 , bn = Fk , which gives the required result for k = n. Similar proof works for Lucas numbers.

Lecture 7 Variations on Catalan Connections and the Children Pairing Game

Catalan numbers are “dense” in combinatorics with hundreds of interpretations in combinatorics, algebra, geometry, and topology. This lecture starts with a simple interpretation of Catalan numbers with a topological flavor and presents a proof, motivated by the theory of skein modules, but which is very elementary and looks like a child’s game. We also discuss the lattice path and Dyck path interpretation of Catalan numbers (including Désiré André reflection principle). Furthermore, we introduce Fuss numbers, a generalization of Catalan numbers by Euler’s assistant at St. Petersburg.

7.1 Introduction In this lecture, we describe some combinatorial structures often used in knot theory. They are related with crossingless connection interpretation of Catalan numbers. It is one of hundreds of interpretation of Catalan numbers (see, e.g., [Stan2]). We start by describing the method the first author developed in 1997, while working on the Kauffman bracket skein modules (see Lecture 12), which makes the calculations very easy to visualize [Prz15, PS1].

7.2 Crossingless Connections and Catalan Numbers Catalan numbers appeared for the first time in a question, posed by Euler, concerning the number of triangulations of a polygon. For us, the most interesting interpretation (as given in the definition below) is concerned with crossingless connections of points on the boundary of the unit disk. It is not clear who first formulated this © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_7

143

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7 Variations on Catalan Connections and the Children Pairing Game

problem. The oldest reference, known to us, is only 90 years old [Err], but we think this example is much older and comes from the time when Euler and his assistant, Fuss, were in St. Petersburg.1 Definition 7.2.1 Consider 2n points on the unit circle (boundary of the unit disk). Let us connect these points by non-crossing arcs. Such connections are called crossingless connections or Catalan states. The number of such Catalan states is called a Catalan number and is denoted by .Cn . Figure 7.1 illustrates all connections for .n = 1, 2 or 3. In particular, .C1 = 1, .C2 = 2, and .C3 = 5.

n=1

n=2

n=3

Fig. 7.1: Catalan states for .n = 1, 2, 3 Our goal is to find a closed formula for .Cn without knowing the answer in advance. We will present a proof found in 1997 when the first author was working on the theory of Kauffman bracket skein modules and, in fact, teaching a graduate class on the topic; [Prz15]. The proof is absolutely elementary and does not require any guesswork about the result and allows many interesting generalizations. See, for example, [PS1]. Thus, we first present the proof when the final result is not known yet and the proof leads to our main result, Theorem 7.2.2. Proof The main idea of the proof is to first consider crossingless connections in an annulus (a disk with a hole). At first, the new problem seems to be more difficult than the original one as we have more connections (we can omit the hole on the left or on the right (see Fig. 7.2). However, the hole gives an orientation to the connections. Let . Dn denote the number of crossingless connections in the annulus with 2n points on the outer boundary. For better visualization (and to have a playful proof), we can imagine that there is a round room with 2n people standing against the wall and a table is placed in the center of the room. Then the people are asked to shake hands in pairs without crossing arms and without shaking hands above the table. 1 Nicolaus Fuss (1755–1826); when Leonard Euler, after an eye operation in 1772, became almost completely blind, he asked Daniel Bernoulli in Basel to send a young assistant, well trained in mathematics, to him in St. Petersburg. It was Nicolaus Fuss who accepted this plea and he arrived in St. Petersburg in May 1773 [DuP, Ozh].

7.2 Crossingless Connections and Catalan Numbers

145

n=1

n=2

Fig. 7.2: Crossingless connections in an annulus for .n = 1, 2

Step 1. If people are hand-connected (we have n connections) and the room is cut into .n + 1 pieces. Thus, there are .n + 1 ways to place a table. This simple reasoning gives . Dn = (n + 1)Cn . Step 2. We now find . Dn . We have to use the table and the first remark is that if the outer circle of the annulus (wall of the room) is oriented, say, positively (i.e., anticlockwise), then every connection, missing the table, is directed (with its orientation agreeing with the orientation of the circle). In particular, every crossingless connection in the annulus (2n)gives a unique choice of n points out of 2n. To show that . Dn is actually equal to . n , we should show the inverse construction, that is, starting from n chosen points, we find connections so that our points are the starting points of crossingless arcs (hands) of connections. We give the algorithm in the form of children playing a game: Assume there are 2n children standing near the rim of the disk (room) of which n are girls and n are boys. Boys have to give their hands to girls without intersections and omitting the table. In the first round of the play, boys look to the right along the wall and if a boy has a girl as a neighbor, he gives the girl his hand (at least one pair is created). The paired children are now out of the game and the game is repeated till all children are paired (the game has to end in at most n steps since in each round one pair is created: a boy gives his hand to the girl directly to his right). From this we conclude the following: Theorem 7.2.2 ( ) (a) . Dn = 2n n (b) .Cn =

1 n+1 Dn

=

( ) 1 2n n+1 n

o

.

The method we just described has many generalizations which can be used, for example, in skein modules (see Lecture 12) and Gram determinants (see Lectures 17 and 18). We discuss some of these generalizations in this lecture.2 We start from the fairly general observation dealing with only partial connections of the points on the boundary. 2 Later in this section, we discuss some other related problems solved by our method. In modern parlance, it would be called “a proof mining”.

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7 Variations on Catalan Connections and the Children Pairing Game

Theorem 7.2.3 Consider a unit disk . D2 with a table inside and N points on its boundary. Let .Ann = (D2 − table) denote the annulus. Consider the set . B(N, p) of crossingless connections of 2p points, .2p ≤ N with the rest of( the ) points connected to the table (again without any crossings). Then . B(N, p) has . Np elements. Proof We follow the proof of Theorem 7.2.2. If a connection is given, then every arc connecting points on the boundary is directed and has an initial point. Thus, we have p initial points. Conversely, if p points are given, we construct connections in the same manner as in Theorem 7.2.2 using the children game. Notice that points not connected in the game (if .2p < N) are not isolated from the table and so can be connected to it. o Our reasoning can be extended to many other instances. We describe below some of these as illustrations. We can, as an example, filter the crossingless connections of 2n points by the number of times they cut a line segment F, which connects the table with the wall (see Figs. 7.3 and 7.4).

b x

x1

2n

F

x2

x

xn

2n−1

x n+1

Fig. 7.3: Table connected to the boundary of the disk

k=0

k=1

k=2

Fig. 7.4: Stratification of crossingless connections in an annulus

7.2 Crossingless Connections and Catalan Numbers

147

Corollary 7.2.4 Consider 2n points, . x1, x2, . . . , x2n , on the boundary of the disk and a base point b between . x2n and . x1 . We connect the table with b by an arc F (as in Fig. 7.3). Let . Dn,k denote the number of states (crossingless connections) in the annulus, in which the number of crossingless arcs cut the interval F exactly k times. Thus, the number of cups disjoint from F is . p = n − k. Similarly, let . Dn, ≥k denote the number of states cutting F at least k times (thus, . Dn,k = Dn, ≥k − Dn, ≥k+1 ). ( ) 2n .Then Dn, ≥k = and as a consequence, n−k ( ) ( ) ( ) ( ) 2n 2n 2n 2n . Dn,k = − = − . n−k n−k −1 p p−1 In particular for . k = 0 we get the Catalan number .

(2n) n

−

( 2n ) n−1

=

( ) 1 2n n+1 n .

Proof We use the same idea as in the proof of Theorem 7.2.2. The game would start with p boys (. p ≤ n) on the rim of the room and they look right to give a hand to a partner who is not a boy and the connected pairs are out of the game. o From the same method, we can draw the following important conclusion about the number of crossingless connections in the Möbius band . Corollary 7.2.5 Replace the table with a crosscap, that ( )is, the setting is changed from the annulus to the Möbius band, and then there are . 2n k crossingless connections in the Möbius band that cut the crosscap exactly .n − k times. Proof We modify the proof of Theorem 7.2.3 by replacing the table with the crosscap. o Exercise 7.2.6 Consider 2n points on the boundary of the Möbius band. How many crossingless connections are there? Hint: Consult Fig. 7.5. Corollary 7.2.7 Consider N points, . x1, . . . , x N (N not necessarily even) on the boundary of the disk . D2 ordered anticlockwise. Let b be a point (or rather a short arc) on .∂D2 between . x N and . x1 ; see Fig. 7.6. We consider partial crossingless connections (using . p ≤ N2 arcs) so that the non-connected . h = N − 2p points could be connected to b without intersections. Then, the number of such connections is: ) ( ( ) ( ) ( ) N N N N (h) .C N = p − p − 1 = N −h − N −h − 1 . 2 2 The same combinatorial problem has been discussed by Di Francesco [DiF] in his work on Gram determinants of semimeanders (see Lecture 17). We now count crossingless connections related to semimeanders. Consider all h to N tangles

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7 Variations on Catalan Connections and the Children Pairing Game

Fig. 7.5: Stratification of crossingless in the Möbius band; ) ( ) (4) (4connections + 1 + 40 . .11 = 2 (4) (4) . 2 = 6 is the number of states that do not have intersections with the crosscap, . 1 is( the ) number in which exactly one arc passes through the crosscap, and . 40 is the unique state in which the arcs pass through the crosscap twice

1 2

···

Fig. 7.6: N points with k connections to the arc b

(.(h, N)-tangles), . h ≤ N without intersection and trivial components such that the points on the top have no connections among themselves. The case when . h = 3, N +h is illustrated in Fig. 7.7. Therefore, the number of .(h, N). N = 5, and .n = 2 (h) semimeanders, that is, .(h, N)-tangles as described above, .CN is as follows: (h) . |C N |

( =

N N −h 2

)

( −

)

N N −h 2

−1

.

Fig. 7.7: .(3, 5) semimeander tangles

7.2 Crossingless Connections and Catalan Numbers

149

Sketch of the Proof: Choose h points from the N points at the bottom of the tangle. By the children game, interpret subsequent connections as elements of .(h ', N)-tangle with . h ' ≥ h. Corollary 7.2.8 Consider 2n points on the boundary of the disk and a chord S which cuts the disk so that boundary points are divided into two groups of sizes .m1 and .m2 = 2n − m1 , respectively (see Fig. 7.8). Then, the number of Catalan connections which cut the chord exactly s times (.s ≤ min(m1, m2 ) and .mi − s is even) is equal to: )) (( )) (( ) ( ) ( m2 m−1 m−1 m2 . − − . m2 −s m2 −s m1 −s m1 −s 2 2 −1 2 2 −1 If we denote by . p1 = m12−s the number of arcs on the left side of the chord, and by m2 −s . p2 = 2 , the number of arcs on the right side, then we get: (( .

)) (( ) ( )) ) ( m2 m1 m1 m2 . − − p1 p1 − 1 p2 p2 − 1

Proof We can shrink the chord S and use Corollary 7.2.7 twice. See Fig. 7.8. ([Prz25] page 112). o

ym

2

ym

n oi

sp

x1

2

ts

x1 b y1

xm

y1 1

x m1

Fig. 7.8: Disk with a chord S cut by a connection s times Another variation of the problem, solved by the same method, is used in the paper by the first author and Qi Chen [ChP2]: Corollary 7.2.9 Consider .2n + 2k points on the boundary of a unit disk, in anticlockwise (positive) order: .a1,2 , . . . , a2n, l1, . . . , lk , uk , . . . , u1 . Consider crossingless connections between these points with the conditions that no points .li are connected to .l j and no points .ui are ˜ connected to .u j , .i = j. Then the number of these connections, denoted by . | NC(D n,k )|, (2n) ( 2n ) is equal to . n − n−k−1 . Proof We propose two proofs of this corollary: 1. We apply Corollary 7.2.7 k times. This is developed and explained in more detail in the proof of Corollary 7.2.4.

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7 Variations on Catalan Connections and the Children Pairing Game

2. In [ChP2], the corollary is proved using Corollary 5.4 as follows: consider 2n points on the boundary of a unit disk with a table inside. Let the table be connected to a point on the boundary between . x2n and . x1 by an arc F. Then ( 2nthe ) number of crossingless connections which cut F at least .n − k times . Thus, the number of connections which cut F at most k times is is . n−k (2n) ( 2n ) − . n n−k−1 . We show that there is a bijection between these connections and ˜ elements of . NC(D n,k ). The trick is to cut a disk along the curve F (see [ChP2] for .o details). Notice that the proof can be used to illustrate the following identities: 1. 2. 3.

.

(2n)

n (2n) . n (2n) . n

− − −

( 2n ) n−1

= Cn ,

n−2

= Cn+1 , and

n−3

= Cn+2 − 2Cn+1 + Cn .

( 2n ) ( 2n )

We can generalize Corollary 7.2.9 slightly to get the following result: Corollary 7.2.10 Consider . N + k1 + k2 = 2n points on the boundary of a unit disk, in anticlockwise (positive) order: .a1,2 , . . . , a N , lk1 , , . . . , l1, u1, . . . , uk2 , with the conditions that no point of .li is connected to .l j and no point of .ui is connected to .u j . Then the number of these connec˜ tions, denoted by . | NC(D N,k1,k2 )|, is equal to ) ( ( ) N N . N − |k −k | − N −k −k . 1 2 1 2 −1 2 2 Proof We know from Corollary .5.7 that if we do not allow connections between ( N ) ( ) N the points .li and .u j , then we get . N −k1 −k2 − N −k1 −k2 −1 possible connections. If we 2 2 connect .l1 with .u1 , with there being no other connections between .l j and .ui , then we ( ) ( ) N N get . N −k1 −k2 −1 − N −k1 −k2 −2 connections. We continue like this, till we connect .l1 2

2

with .u1 , .l2 with .u2 ,. . . , and finally .lmin(k1,k2 ) with .umin(k1,k2 ). The last situation gives ( N ) ( ) ( N ) ( ) N N . N −|k1 −k2 | − N −|k1 −k2 | connections. Thus, in total we get . N −|k1 −k2 | − N −k1 −k2 −1 −1 2 2 2 2 connections, as predicted. o We can further generalize Corollary 7.2.10 and we leave this as the following exercise for the reader. Exercise 7.2.11 Consider .2n = N + k1 + . . . + k m points on the boundary of the unique disk. Find the formula for .CN,k1,..,kn of crossingless connections such that no points in the group . ki , for any i, are connected.

A different type of generalization of our method considers connecting k-spiders instead of connecting arcs. That is, we consider sn points on the boundary of the

7.2 Crossingless Connections and Catalan Numbers

151

disk connected in groups of s (called spiders with s legs) without intersection. The same method as in the proof of Theorem 7.2.2 works—adding a table allows us to find the first leg for any spider In Fig. 7.9 we draw all 3-spiders with .s = 3 legs and .3n = 6 points on the boundary of the disk. The general formula for the number of s-legged spiders is easily obtained by our method. We leave the exact calculation of this number as an exercise for the reader. n=2, s=3

Fig. 7.9: 3-spiders for .n = 2 ( sn) 1 Exercise 7.2.12 Show that .Fn,s = 1+(s−1)n n . We call the number . Fn,s the Fuss number, as Nicolaus Fuss was the first to consider them [Fus]. Exercise 7.2.13 Consider N points on the boundary of a disk and allow spiders of different sizes; say .ai spiders of i legs, .i ≥ 1. Denote by k the number of spiders k E (see Fig. 7.10). Thus, we have . N = iai . Show that the number of nonintersecting i=1

spider positions is equal to: (

1

.

1+

( Here, .

n E

ni

i=0 n1, n2, . . . , nn

k E i=1

)

)

. k E (i − 1)ai a1, a2, . . . , ak , N − i=1 ai (

=

N

n E i=0

ni )!

n1 !, n2 !, . . . , nn !

is a multinomial coefficient.

Fig. 7.10: Example with . N = 15, .a1 = 2, .a2 (= 3, .a3 )= a4 = 1; thus, four spiders and 15 positions nine regions. We get . 19 2,3,1,1,8

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7 Variations on Catalan Connections and the Children Pairing Game

Hint: The number .1 +

k E i=1

(i − 1)ai counts the number of regions of the disk minus

spiders, and as in the previous exercise, we have to choose, in the room with a table, the first legs of every spider from N points; compare [PS1]. Exercise 7.2.14 Consider 2n points on the boundary of the disk, and let .Cn,(k) denote the number of connections (by arcs) with exactly k crossings. For example, .Cn,(0) = ( ) 1 2n C0 = n+1 n is the Catalan number. (1) Find .C3,(k) and .C4,(k) for any k. (2) Find .C6,(3) and .C7,(3) . (3) Find a closed formula for .Cn,(1) . (4) Find a closed formula for .Cn,(2) . (5) Find a formula for .Cn,(3) . Hint: For . k = 1, place the table in place of the crossing. For . k = 2, generalize Exercise 7.2.12 to spiders of different numbers of legs. For . k = 3 The best formula we know is: ( ) ( ) ( ) 2n 2n 2n 1 .Cn,(3) = + 3n + , n−3 n−4 3 2, n − 6, n + 4 ( 2n ) 2 or, as suggested by Krzysztof Kowitz, . n +2n−9 n−3 . 6 See [Rio] for a general discussion on numbers .Cn,(k) .

7.3 Lattice Path and Dyck Path Interpretations of Catalan Numbers There are hundreds of interpretations and proofs concerning Catalan numbers [Stan3]; here we choose two of them to be used later for the study of Gram determinants (Lectures 17 and 18). Each of the cases in the definition below has Catalan number of elements. The first is immediately equivalent to the case we considered before in Definition 7.2.1 where we had 2n points on the unit circle. Definition 7.3.1 (1) Consider ways of connecting 2n points in the plane on a horizontal line by n nonintersecting arc, each arc connecting two of the points and lying above the points.

7.3 Lattice Path and Dyck Path Interpretations of Catalan Numbers

153

(2) Consider lattice paths from .(0, 0) to .(n, n) with steps .(0, 1) and .(1, 0), never rising above the line . x = y. (3) Consider Dyck paths from .(0, 0) to .(2n, 0), i.e., lattice paths with steps .(1, 1) and .(1, −1), never falling below the x-axis. Exercise 7.3.2 Show that there is a bijection between sets described in (1), (2), and (3); thus, each case has Catalan number of elements. Hint Show that each set can be described by a sequence . x1, x2, . . . x2n where . xi = 1 or .−1, the sum is equal to zero, and all partial sums are nonnegative. In (1) we can think of 1’s as boys and .−1 as girls in the children game. In Fig. 7.11, the three interpretations are given graphically for the sequence bbbggbgg. (4,4)

g b

b g b

g g b

b b g

g

b g

g

(0,0)

b

b

b

g g

b (0,0)

b

g g (8,0)

Fig. 7.11: Interpretations of the sequence bbbggbgg Exercise 7.3.3 Show directly that the number of sequences . x1, x2, . . . x2n where xi = 1 (or ).−1, the sum is equal to zero, and all partial sums are nonnegative is equal 1 2n to . n+1 n .

.

Hint: Use Désiré André reflection principle or more precisely involution in ballot ( ) sequence [AD]). More precisely, without restriction on partial sum, we have . 2n n sequences. If a sequence . x1, x2, . . . x2n is negativeEfor some partial sum (“bad” k xi = −1. Consider partial sequence), then let k be the smallest number with . i=1 reflection on the sequence that is the sequence .−x1, . . . , −xk , xk+1, . . . , xn . The sum of the elements of such ( 2n a) sequence is equal to 2 as we have two more 1’s than .−1. such sequences and the same number of “bad” sequences Obviously, there are . n−1 . x1, x2, . . . x2n . Thus, the number of “good” sequences, that is, with nonnegative ( ) ( 2n ) ( ) 1 2n partial sums, is equal to . 2n n − n−1 = n+1 n . The realization of this idea in the lattice paths’ example of Definition 7.3.1 is called the Désiré André reflection principle. We will show another use of this principle in Lectures 17 and 18.

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7 Variations on Catalan Connections and the Children Pairing Game

7.4 Exercises Exercise 7.4.1 Illustrate all distinct Catalan states (crossingless connections in the disk) for n = 4. Exercise 7.4.2 Illustrate all distinct crossingless connections in the Möbius band between six points on the boundary that intersects the crosscap exactly once. Explain the connection between the number of distinct states to Theorem 7.2.3. Exercise 7.4.3 Prove the following identities: 1. 2. 3.

(2n) n (2n) n (2n) n

− − −

( 2n ) n−1

= Cn ,

n−2

= Cn+1 , and

n−3

= Cn+2 − 2Cn+1 + Cn .

( 2n ) ( 2n )

Exercise 7.4.4 Illustrate the lattice path from (0, 0) to (4, 4), Dyck path from (0, 0) to (8, 0), and the connection between eight points in the plane on the horizontal line by four nonintersecting arcs such that each illustration corresponds to the given sequences. (i) {1, −1, 1, 1, −1, −1, 1, −1}. (ii) {1, 1, −1, 1, −1, 1, −1, −1}. (iii) {1, 1, 1, 1, −1, −1, −1, −1}. Exercise 7.4.5 Illustrate all possible lattice paths from (0, 0) to (3, 3).

Lecture 8 The Temperley-Lieb Algebra and the Artin Braid Group

The Temperley-Lieb algebra was originally described in the work of two physicists; Temperley and Lieb. However, this algebra has deep connections to knot theory and 3-manifold invariants. For example, its connection to knot theory stems from Louis H. Kauffman’s interpretation of the Temperley-Lieb algebra as a diagrammatic algebra consisting of n-tangles as its basis. In this lecture we explore the basics of the Temperley-Lieb algebra and Artin braid group.

8.1 Introduction The first formal definition of the Temperley-Lieb algebra was given by Rodney J. Baxter in [Bax] while describing the work of physicists Neville Temperley and Elliott Lieb [TL]. Vaughan Frederick Randal Jones introduced the algebra independently while working on von Neumann algebras in [Jon1]. Around 1983 he was alerted by David E. Evans that the algebra he described in [Jon1] is equivalent to the TemperleyLieb algebra discussed in [Bax]. In [Jon1] Jones introduced a normal form for the Temperley-Lieb algebra. Louis Kauffman constructed a graphical interpretation [Kau6] for this algebra also described in this lecture. While the algebra can be considered over .Z[d], in many cases, it is considered over .C or the field of rational functions in the variable d, .Q(d), whose elements are functions of the form .P/Q where P, .Q ∈ Z[d]. Kauffman, motivated by his Kauffman bracket, considers the algebra over .Z[A±1 ] where A is an indeterminate and .d = −A2 − A−2 . However, the Temperley-Lieb algebra can be defined over an arbitrary commutative ring R with unity and an element .d ∈ R. In Lecture 9, we follow Jones’ construction of the Jones-Wenzl idempotent of the Temperley-Lieb algebra. The construction requires some basic knowledge of braids and an epimorphism from the Artin braid group onto the permutation group which we will discuss in this lecture. We also need an epimorphism from the group ring over the Artin braid group onto the Temperley-Lieb algebra. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_8

155

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8 The Temperley-Lieb Algebra and the Artin Braid Group

8.2 The Temperley-Lieb Algebra Definition 8.2.1 Let R be a commutative ring with unity and d ∈ R. Let n ∈ N be fixed, then the Temperley-Lieb algebra, denoted by TLn , is defined to be the unital associative algebra over R with generators e1, . . . , en−1 , identity element 1n , and relations: 1. ei e j ei = ei for |i − j | = 1. 2. ei e j = e j ei for |i − j | > 1. 3. ei2 = dei . Lemma 8.2.2 ([Jon1]) Any word in the generators e1, . . . , en−1 of TLn can be written as a product of a power of d and a product of the generators such that the largest index appears only once. Proof We will prove this using a double induction following Christian Kassel and Vladimir Turaev in [KT]. The double induction will be on the pair (m, k) in lexicographical order (i.e., (m, k) < (m ', k ' ) if and only if either m < m ' or m = m ' and k < k ') where m is the largest index of a generator, em , in the word and k is the number of times the element of the highest index occurs in the product. The initial cases are pairs (m, 1) for which the theorem follows from the definition. Let E be a word in the generators, m be the highest index in E, and k be the number of times em occurs in E. Assume by induction that the lemma holds for (m ', k ') < (m, k). We write E in the following way: .

E = E1 em E ' em E2,

where E1 and E2 may contain em in their product but E ' does not. We can break this down into two cases, either the largest index of E ' is less than m − 1 or it is equal to m − 1. Case 1: Suppose the largest index of E ' is less than m − 1, then by relation (2), .

E = E1 e2m E ' E2 = dE1 em E ' E2 .

This reduces the number of em s to k − 1 and by the inductive hypothesis we are done. Case 2: Suppose the largest index of E ' is equal to m − 1. Then by the inductive hypothesis E ' can be rewritten as E ' = d l E3 em−1 E4 , where E3 and E4 do not contain em−1 . In fact the highest index in their product is less than m − 1 . Therefore, by relation (2) then relation (1), .

E = E1 em d l E3 em−1 E4 em E2 = d l E1 E3 em em−1 em E4 E2 = d l E1 E3 em E4 E2 .

This reduces the number of em s to k − 1 and by the inductive hypothesis we are done. o

8.2 The Temperley-Lieb Algebra

157

Theorem 8.2.3 (Normal Form) [Jon1] Any word in the generators e1, . . . , en−1 of TLn can be uniquely written as a product of a power of d and an element in normal form as defined below. Let is ≥ js, 0 < i1 < · · · < ik < n, and 0 < j1 < · · · < jk < n, then the following element is in normal form: .

Ei1, j1 Ei2, j2 · · · Eik , jk , where Eis , js = eis eis −1 . . . e js .

th The number ( ) of distinct elements in normal form is the n Catalan number, 1 2n Cn = n+1 n .

Proof For each element in normal form we associate to it a lattice path from (0, 0) to (n, n) in the following way: .

(0, 0) → (i1, 0) → (i1, j1 ) → (i2, j1 ) → (i2, j2 ) → (i3, j2 ) → (i3, j3 ) → · · · · · · → (ik , jk−1 ) → (ik , jk ) → (n, jk ) → (n, n).

(8.1)

Such paths with the conditions is ≥ js, 0 < i1 < · · · < ik < n, and 0 < j1 < · · · < jk < n are all lattice paths from (0, 0) to (n, n) such that they are below the diagonal which means there are nth Catalan number of paths associated to elements in normal form. See Lecture 7 for more details on Cn . Let E be a word in the generators of TLn . We will prove this by induction on the length of the word. Let l be the length of the word E, for the base case l = 1, E = ek = Ek,k for some k and we are done. Assume by induction that the theorem holds for words of length less than l. Suppose m is the largest index in E. Then by Lemma 8.2.2, E can be written as .

E = d μ E1 em E2,

where μ ≥ 0 is an integer and the largest indices of E1 and E2 are less than or equal to m − 1 (E1 or E2 can be empty). We will consider two cases. Case 1: Suppose the highest index of E2 is less than m − 1. Then by relation (2) E = E1 E2 em . Since the length of E1 E2 is less than l, then by the inductive hypothesis we are done. Case 2: Suppose the highest index of E2 is equal to m − 1. Since the length of E2 is less than l, then by the inductive hypothesis E2 = d μ2 Ei1, j1 Ei2, j2 · · · Eik , jk for is ≥ js, 0 < i1 < · · · < ik < n, and 0 < j1 < · · · < jk < n where ik = m − 1. As we push em to the right as far as possible, we obtain, .

em E2 = d μ2 Ei1, j1 Ei2, j2 · · · Em, jk .

Now we have written E in such a way that .

E = d μ+μ2 E1 Ei1, j1 Ei2, j2 · · · Em, jk ,

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8 The Temperley-Lieb Algebra and the Artin Braid Group

where E1 Ei1, j1 Ei1, j1 · · · Eik−1, jk−1 is a word in the generators of TLn with the highest index less than m and length less than l. By the inductive hypothesis we can rewrite the word in normal form, that is, E1 Ei1, j1 Ei1, j1 · · · Eik−1, jk−1 = d μ1 Ei1' , j1' Ei2' , j2' · · · Eik' ' , jk' ' where is' ≥ js', 0 < i1' < · · · < ik' ' < n, 0 < j1' < · · · < jk' ' < n, and ik' ' ≤ m − 1. Therefore, we have .

E = d μ+μ2 +μ1 Ei1' , j1' Ei2' , j2' · · · Eik' ' , jk' ' Em, jk .

(8.2)

We have two more cases to consider. Case 2A: Suppose jk' ' < jk , then we have constructed a normal form. Case 2B: Suppose jk' ' ≥ jk , then push e jk' ' to the right as far as possible to obtain the subword e jk' ' e jk' ' +1 e jk' ' which is equal to e jk' ' . Thus, we have written E as a word of length less than l and by the inductive hypothesis we are done. o Remark 8.2.4 Notice that our proof of Lemma 8.2.2 and Theorem 8.2.3 is algorithmic and in fact it is a “greedy” algorithm, in the sense that on the path from a given word to its normal form we never increase the length of the word.1

8.3 Temperley-Lieb Algebra’s Connections to Tangles John H. Conway introduced the concept of a tangle while working on rational knots. We use this notion to describe Kauffman’s graphical interpretation of .TLn . Definition 8.3.1 An .n-tangle is a part of a link diagram with 2n boundary points. The diagram is taken up to ambient isotopy modulo boundary points. In this lecture we will consider an n-tangle as a rectangular shaped disk with n boundary points on the left and n boundary points on the right, calling them the input and output points, respectively. Kauffman interpreted the Temperley-Lieb algebra starting from the basis of crossingless connections. The identity element corresponds to an n-tangle with n parallel strands where each ith input is connected to the ith output, shown in Fig. 8.1a. Each .ei corresponds to an n-tangle that has one input and one output cap on the ith and .i + 1th position, as shown in Fig. 8.1b. Multiplication2 is defined by combining tangles from left to right and corresponds to identifying the right side of the first n-tangle to the left side of the second ntangle while respecting the boundary points; see Example 8.3.2. Notice that after multiplication it is possible to obtain homotopically trivial curves, which we denote by d; see Example 8.3.3. Throughout this lecture, any strand labeled with a number, 1 For the formal definition of a greedy algorithm, see [CLRS]. 2 Many authors denote tangles in the vertical position and in this way multiplication is defined by identifying tangles from top to bottom.

8.3 Temperley-Lieb Algebra’s Connections to Tangles

159

say n, will express n parallel strands, and if a strand is not marked with a number, it will denote a single strand. −1 • • •

= − −1

(a) Identity element:

parallel strands.

(b)

.

Fig. 8.1: The generators of .TLn

Example 8.3.2 For .n = 4, multiplication of .e1 and .e2 is diagrammatically constructed using the diagrams associated to .e1 and .e2 .

Since

1

=

and

1 2

2

=

, then

=

=

.

.

Example 8.3.3 Diagrammatically, it is possible to obtain homotopically trivial curves after multiplication of two elements in the basis; each time this happens, assign each homotopically trivial curves by the element .d ∈ R. 2 2

=

=

=

2

Definition 8.3.4 The n-tangle algebra is an R-module with basis n-tangles and multiplication described above. Example 8.3.5 For .n = 4, consider the following three tangles:

We have the following relation:

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8 The Temperley-Lieb Algebra and the Artin Braid Group

1 2 1 .

=

=

=

1,

=

,

2 1 2

=

=

=

2,

and 1 3

=

=

=

=

3 1.

Definition 8.3.6 The diagrammatic algebra is a subalgebra of the n-tangle algebra generated by tangles without crossings. Example 8.3.7 As an n-tangle, .Eis , js = eis eis −1 · · · e js for .is > js is represented by −1

−1 ••

−

•

=

− −

−1

+2

−1

.

Lemma 8.3.8 All crossingless connections on 2n boundary points with caps can be decomposed into a product of .e1, e2, . . . , en−1 . Proof Fix n. Let D be a diagram with k input caps and assume that the input cap on the ith position is the minimal input cap, that is, for any input cap on the lth position, where .l = i, we must have .i < l. In order to have a crossingless connection, there must be exactly k output caps. Suppose the output cap occurs on the jth position and let .m = |i − j | be the distance of the input and output caps. For . k = 1 and .m = 0, −1

=

= − −1

.

8.3 Temperley-Lieb Algebra’s Connections to Tangles

161

This is our base case for m. For . k = 1, assume by induction that for .m − 1 a diagram can be decomposed into a product of generators. For .m > 0, there are two cases to consider. Case 1A: Suppose .i < j. Then we may stretch every strand except the ith input cap over to the right and then deform the .i + 2th strand to obtain,

.

where . D ' has input output caps of distance .(i − 1) − j = (i − j) − 1 = m − 1. By our inductive hypothesis we are done. Case 2A: Suppose .i > j. Then in a similar fashion as case 1A we obtain,

.

,

where . D ' has input and output caps of distance .m − 1. By our inductive hypothesis we are done. Suppose . k > 1. Assume by induction that a diagram with . k − 1 input caps can be decomposed into a product of generators. Additionally assume that the input cap on the ith position is the minimal input cap. Then there exists an output cap on the jth position with the following three cases: Case 1B: If . j = i, then we may stretch every strand except the ith input cap over to the right and stretch the ith output cap over to the left. We may do this because there are no curves between the ith input and output caps. This is because D is a crossingless connection. After such a deformation we obtain,

.

,

162

8 The Temperley-Lieb Algebra and the Artin Braid Group

where . D1 is a diagram that falls in the base case . k = 1 and .m = 0 and . D2 is a diagram with . k − 1 input caps. We have shown that a diagram with .m = 0 and . k = 1 is just .ei , and by the inductive hypothesis . D2 can be decomposed into a product of generators. Case 2B: If .i < j, then for each strand, s, such that .s < i or .s > j + 1 we stretch s to the right and for s such that .i ≤ s ≤ j + 1 we stretch the strand to the left. By performing such deformations we obtain,

,

.

where . D1 is a diagram that falls in case 1A and . D2 is a diagram with . k − 1 input caps. By case 1A . D1 can be decomposed into a product of generators, and by the inductive hypothesis . D2 can also be decomposed into a product of generators. Case 3B: If .i > j, then for each strand, s, such that .s < j or .s > i + 1 we stretch s to the right and for s such that . j ≤ s ≤ i + 1 we stretch the strand to the left. By performing such deformations we obtain,

.

where . D1 is a diagram that falls in case 2A and . D2 is a diagram with . k − 1 input caps. By case 2A, . D1 can be decomposed into a product of generators, and by the o inductive hypothesis . D2 can also be decomposed into a product of generators. Theorem 8.3.9 ([Kau6]) The diagrammatic algebra described above is isomorphic to .TLn and can be thought of as a diagrammatic interpretation of it. Proof For the purpose of this proof, we denote the graphical interpretation of .TLn -n sending .ei to .e-i . -n and the generators by .e-i . Consider the map .g : TLn → TL by .TL Since .e-i satisfies the Temperley-Lieb relations (compare Exercise 8.5.1), g is a welldefined epimorphism of algebras. To prove monomorphism we use the following -n is a free module with Catalan number of elements in its lemma and the fact that .TL basis. o Lemma 8.3.10 The set of all elements in normal form provides a basis of the Temperley-Lieb module .TLn .

8.4 The Artin Braid Group

163

Proof Since .TLn has Catalan number of normal form elements and g is an epimoro phism, then elements in normal form provide a basis for .TLn .

8.4 The Artin Braid Group In 1923, James W. Alexander in [Ale2] made a significant connection between braids and links by proving that any link can be written as a closure of a braid. However, the first deep study of braid groups was conducted by Emil Artin in [Art1, Art2, Art3]. In 1936, Andrei Andreyevich Markov Jr. found moves on the closure of braids that are similar to the Reidemeister moves that produces ambient isotopic links.3 In this subsection we will discuss braids and their relationship to links. Definition 8.4.1 The Artin braid group is defined by the following group presentation: .

Bn =< σ1, . . . , σn−1 ; σi σj = σj σi for |i − j | > 1, σi±1 σi σi±1 = σi σi±1 σi > .

The Artin braid group can be interpreted using n-tangles where elements of . Bn are positive braids. That is, an element in . Bn is an n-tangle with positive crossings such that when read from left to right, each generator .σi corresponds to the positive crossing of the .i th and .i + 1th strands. That is, the .i th generator element .σi is a positive transposition of the .i th and .i + 1th strands. Example 8.4.2 For .n = 3, the braid relation .σ2 σ1 σ2 = σ1 σ2 σ1 can be realized by the third Reidemeister move. 2 1 2 .

=

3

← →

=

2 1 2

.

Definition 8.4.3 Let . B ∈ Bn , then the closure of a braid B is obtained from B by identifying the ith input with the ith output; the result is a link diagram. Alexander in [Ale2] was the first to observe that links can be represented as the closure of braids. Theorem 8.4.4 (Alexander’s Theorem) [Ale2] Every link can be represented as a closure of a braid. Example 8.4.5 Consider the braid word .(σ1 σ2 σ3 )3 ∈ B4 , the braid representing this braid word is shown in Fig. 8.2a, and its closure is the torus knot of type .(4, 3) as shown in Fig. 8.2b. 3 In fact, Markov was using one additional move which known as an exchange move but Noi Moisevich Weinberg (1914–1942) in [Wei] showed in 1939 that this move is not needed.

164

8 The Temperley-Lieb Algebra and the Artin Braid Group

Fig. 8.2: The closure of the braid representing .(σ1 σ2 σ3 )3 ∈ B4 An example of the non-uniqueness of the representation of links as a closure of a braid is illustrated in Fig. 8.3. Markov in [Mar] introduced moves on the closure of braids that produce ambient isotopic knots; these moves are known as Markov moves.

(a) Closure of braid 1 ∈ 2

(b) Closure of braid

−1 1

∈

2.

(c) Closure of braid 1 ∈

1.

Fig. 8.3: The closure of the braids representing .σ1, σ −1 ∈ B2 and .1 ∈ B1 are all ambient isotopic to the unknot

Exercise 8.4.6 Find two examples of distinct braid words whose closure is ambient isotopic to the torus knot of type .(4, 3) but is distinct from .(σ1 σ2 σ3 )3 ∈ B4 , as shown in Fig. 8.2a. Definition 8.4.7 There are two types of Markov moves4 plus braid isotopy on the closure of braids: stabilization and destabilization. The moves are supported by a braid axis fixed in the middle of the closure that intersects the projection plane once as shown on Fig. 8.4. Braid isotopy is the isotopy of the closure of the braid supported in the complement of the braid axis. That is, braid isotopy of the closure of the braid is isotopy that cannot pass through the braid axis. Stabilization is the process of performing a Reidemeister I move by adding a crossing and pushing the loop across the braid axis once, as shown in Figs. 8.5 and 8.6. Destabilization is the inverse of stabilization. 4 On the algebraic level we only need conjugation, stabilization, and destabilization. See [Mor1].

8.4 The Artin Braid Group

165

−−−−−→ closure

Fig. 8.4: Closure of braid with braid axis

Fig. 8.5: A positive stabilization is a Reidemeister I move which crosses the braid axis and introduces a positive crossing to the braid and increases the index

Fig. 8.6: A negative stabilization is a Reidemeister I move which crosses the braid axis and introduces a negative crossing to the braid and increases the index

Theorem 8.4.8 (Markov’s Theorem) Two braids . B1, B2 have isotopic closures in S 3 if and only of we can obtain . B1 from . B2 by a sequence of Markov moves.

.

For a proof of Theorem 8.4.8 we refer the reader to [BC, LR, Trac]. An epimorphism from the Artin braid group to .Sn , . p : Bn → Sn , is defined by sending generators of . Bn , .σi , to the transpositions in .Sn ; .si = (i, i+1) for .1 ≤ i ≤ n−1. Recall that the permutation group .Sn has the following group presentation: .

Sn =< s1, . . . , sn−1 ; si2 = 1, si s j = s j si for |i − j | > 1, si±1 si si±1 = si si±1 si > .

By using this epimorphism we can now uniquely interpret a braid word using elements of .Sn . For a permutation .π ∈ Sn , let .bπ denote the unique minimal positive braid word such that . p(bπ ) = π. This epimorphism is crucial to defining the JonesWenzl idempotent of the Temperley-Lieb algebra, which will be discussed in the next lecture. We also obtain a subgroup of the Artin braid group from this epimorphism, as described below.

166

8 The Temperley-Lieb Algebra and the Artin Braid Group

Definition 8.4.9 The pure braid group, denoted by .PBn , is defined to be the kernel of the epimorphism . p : Bn → Sn (the set of elements in . Bn that maps to the trivial permutation). Generators of the pure braid group are of the form .

2 −1 −1 −1 σi j = σi σi+1 · · · σj−2 σj−1 σj−2 · · · σi+1 σi .

Diagrammatically, elements of .PBn are tangles such that each strand starts and ends in the ith position for .1 ≤ i ≤ n − 1. Example 8.4.10 Consider .PB4 , a generator for this pure braid is .σ2,4 , and its diagrammatic braid representations are shown in Eq. 8.3.

(8.3) Example 8.4.11 Due to the first relation in the presentation of .Sn (.si2 = 1), elements can be expressed in various ways. For example, in .S3 , .(12) is equivalent to .(23)(23)(12); they both represent the same element in . S3 . Notice that

and

.

This is an example of distinct braids with equivalent images under p. In this example the first braid is the unique minimal positive braid word that corresponds to .(12) ∈ S3 ; it is the braid word with minimal positive crossings such that the image is equal to .(12). Theorem 8.4.12 ([Jon1]) The minimal positive braid word is unique for a given permutation. Theorem 8.4.13 ([KL]) The algebra of n-tangles modulo the Kauffman bracket relations is isomorphic to .TLn . As a module the algebra of n-tangles is a special case of the relative Kauffman bracket skein module of the surface cross the interval; see Lecture 13. Theorem 8.4.13 follows from Theorem 13.5.2. Exercise 8.4.14 (a) Compute the image in .TL3 of the 3-braid .Δ = σ1 σ2 σ1 representing a positive half twist on 3-strands. (b) Compute the image .TL3 of the 3-braid .σ1 σ22 σ1 .

8.5 Exercises

167

8.5 Exercises Exercise 8.5.1 Prove ei e j = e j ei for |i − j | > 1. Exercise 8.5.2 Make a 2 × 2 multiplication table for TL2 . Exercise 8.5.3 Make a 5 × 5 multiplication table for TL3 . Exercise 8.5.4 Compute the image in TL3 of the 3-braid (σ1 σ2 )3 representing a positive full-twist on 3-strands. Exercise 8.5.5 Compute the image in TL4 of the 4-braid (σ1 σ2 σ3 )4 representing a positive full-twist on 4-strands. Exercise 8.5.6 Find a braid word in B3 whose closure is the positive trefoil knot 31 .

Lecture 9 Symmetrizers of Finite Groups and the Jones-Wenzl Idempotent

The Jones-Wenzl idempotent, an element of the Temperley-Lieb algebra, has several interesting properties. In particular, it is a useful tool for simplifying computations and proving theorems in knot theory. In this lecture, we introduce symmetrizers of finite groups then present the Jones-Wenzl idempotent as a certain symmetrizer. The lecture also describes properties associated to the Jones-Wenzl idempotent— specifically the recursive definition given by Hans Wenzl.

9.1 Introduction The Jones-Wenzl idempotent is an idempotent element in the Temperley-Lieb algebra. This was found by Vaughan Frederick Randal Jones as a certain symmetrizer (see Definition 9.3.7 and [Jon1]) using the braid group and the projection to an algebra which was later recognized as isomorphic to the Temperley-Lieb algebra as described by Rodney J. Baxter in [Bax]. Later, Hans Wenzl found a recursive formula for this idempotent (see Theorem 9.3.16 and [Wen]). In this lecture, we discuss finite group symmetrizers and Jones’ generalization to the Artin braid group. In the following lectures, we will see applications of the Jones-Wenzl idempotent, for example, in Lecture 16 we will show how the Jones-Wenzl idempotent is used to prove the existence of the Witten-Reshetikhin-Turaev invariant and how it is used to simplify computations of the invariant. In Lecture 17 we present Xuanting Cai’s work in [Cai] where he uses the idempotent to prove a closed formula for the Gram determinant of type A.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_9

169

170

9 Symmetrizers of Finite Groups and the Jones-Wenzl Idempotent

9.2 Symmetrizers of Finite Groups Definition 9.2.1 Let G be a finite group and ZG be a free module over Z with basis G, then the unnormalized symmetrizer, denoted FG , is defined by the following formula: E . FG = g. g ∈G

FG has the following properties: 1. For any g ∈ G, gFG = FG = FG g. 2. FG FG = |G|FG . Property (1) follows from the definition. That is, let g ' ∈ G, then E E ' ' .g FG = g g= g ' g = FG . g ∈G

g ∈G

Notice g ' is just rearranging the elements of G in the summation. Similarly, it can be shown that FG g ' = FG . Exercise 9.2.2 Use property (1) to prove property (2). By using property (2), we define the normalized symmetrizer, denoted fG , such that fG fG = fG . Definition 9.2.3 The normalized symmetrizer (or symmetrizer) is an element in QG (a free module over Q with basis G), defined by the following formula: .

where FG =

E g ∈G

fG =

FG , |G|

g.

We may use property (2) from the unnormalized symmetrizer to show that the symmetrizer is an idempotent element in QG, .

fG fG =

FG FG FG = = fG . |G| |G| |G|

Definition 9.2.4 Let G be a finite group with epimorphism s : G → {−1, 1} = Z2 . Then the unnormalized anti-symmetrizer is an element of ZG defined by E . Fs = s(g)g. g ∈G

9.3 The Jones-Wenzl Idempotent

171

Fs has the following properties: 1. For any g ∈ G, gFs = Fs g = s(g)Fs . 2. Fs FG = 0. 3. Fs Fs = |G|Fs . The normalized anti-symmetrizer is an element of QG and is defined by .

fs =

Fs . |G|

Exercise 9.2.5 Prove properties (1) thru (3) for the unnormalized anti-symmetrizer.

9.3 The Jones-Wenzl Idempotent In [Jon1] Jones generalized the idea of symmetrizers by defining an unnormalized A-symmetrizer. This was done by using an epimorphism . p : Bn → Sn from the Artin Braid group to the permutation group and then a map to the Temperley-Lieb algebra by evaluating the resulting tangles in the Kauffman bracket skein module. In this way, the normalized A-symmetrizer yields the Jones-Wenzl idempotent of the Temperley-Lieb algebra. Definition 9.3.1 Let .Z[A±1 ] denote the ring of Laurent polynomials in the variable A. Define an unnormalized A-symmetrizer .Fn ∈ Z[A±1 ]Bn (a free module over ±1 .Z[A ] with basis . Bn ) given by the following formula: E . Fn = (A3 ) |π | bπ , π ∈Sn

where . |π| denotes the minimal length of the permutation .π written as elementary transposition generators and .bπ denotes the unique minimal positive braid word such that . p(bπ ) = π. . Fn modulo the Kauffman bracket skein relations yield an element in the TemperleyLieb algebra, .TLn , which is the setting we will work in. Example 9.3.2

Evaluating .F2 modulo the Kauffman bracket skein relations yield:

172

9 Symmetrizers of Finite Groups and the Jones-Wenzl Idempotent

.

Notice the coefficient of the identity element is .[2] A4 !, where .[n] A4 = 1 + A4 + A8 + n | | 4n −1 and .[n] A4 ! = [i] A4 as defined in Definition 9.3.3. · · · + A4(n−1) = AA4 −1 i=1

Definition 9.3.3 .[n]q = 1+q+· · ·+q n−1 is called a quantum integer, .[n]q ! =

n | |

[k]q

k=1

is called a q-factorial. Theorem 9.3.4 .Fn ei = ei Fn = 0 for .ei ∈ TLn , where .TLn is the Temperley-Lieb algebra described in Lecture 8 and .Fn is evaluated modulo the Kauffman bracket skein relations . Proof We will start with .n = 2, then use this result to prove the theorem for any n ∈ N.

.

As seen in Example 9.3.2,

2

= (1 +

4

)

+

2

(−

2

+

2

Therefore,

2 1

.

= (1 +

4

)

+

2

= (1 +

4

)

+

2

−

−2

)

=0

Also, 1 2

= (1 +

4

)

=0

Next, we will use .F2 e1 = 0 to generalize the result to any n and any .ei , where 1 ≤ i ≤ n − 1.

.

9.3 The Jones-Wenzl Idempotent

173

If .π ' is the identity element of .Sn and .π '' = si , which is realized by switching the ith and .i + 1th strands, then by the same argument as above: (bπ ' + A3 bπ '' )ei = 0,

.

(9.1)

when evaluated modulo the Kauffman bracket skein relations. Furthermore, we can pair permutations in the following way: let .π ' be any permutation in which the ith and .i + 1th strands never cross each other, which is a generalization of the .π ' described above. We pair this permutation with .π '', which is the permutation obtained from .π ' by switching the ith and .i + 1th strands, namely, '' ' . π = π si . Notice, by construction, we have .

|π '' | = |π ' | + 1.

(9.2)

We can then use Eq. 9.2 and the same argument used to prove Eq. 9.1 to obtain the following result: '

''

((A3 ) |π | bπ ' + (A3 ) |π | bπ '' )ei = 0.

.

o

This yields the desired result. Corollary 9.3.5 Let .π ∈ Sn , then .

Fn bπ = λb π Fn,

where .λb π ∈ Z[A±1 ] is the coefficient of the identity element in the linear combination of .bπ modulo the Kauffman bracket skein relations . Proof Recall, for .π ∈ Sn , .bπ is a braid with all positive crossings on n strands. If we evaluate .bπ modulo the Kauffman bracket skein relations , we obtain a linear combination of elements in .TLn . By Theorem 9.3.4, all elements spanned by products of .ei s for .1 ≤ i ≤ n − 1 are annihilated when multiplied by .Fn . Therefore, only the o identity element and its coefficient, say .λb π , remains. Theorem 9.3.6 .Fn Fn = [n] A4 !Fn . Proof This property is trivially true for .n = 1. For .n ≥ 2, we use the fact that any permutation .π ∈ Sn can be uniquely written as ' ' . π = σi σi−1 . . . σ1 π , where .1 ≤ i ≤ n − 1 and . π ∈ Sn−1, which is generated by .σ2, . . . , σn−1 , to obtain: E . Fn = (A3 ) |π | bπ π ∈Sn

(

=

E π ' ∈Sn−1

) 3 |π ' |

(A )

(

) bπ ' + A3 bσ1 π ' + . . . + A3(n−1) bσn−1 ...σ1 π ' .

174

9 Symmetrizers of Finite Groups and the Jones-Wenzl Idempotent

A diagrammatic depiction of this step is the following:

.

By using the Kauffman bracket skein relations on .σi , for .1 ≤ i ≤ n − 1, and Theorem 9.3.4, .Fn Fn reduces to ( n−1 ) E 4i . Fn Fn = A Fn−1 Fn = [n] A4 Fn−1 Fn . i=0

The inductive assumption .Fn−1 Fn−1 = [n − 1] A4 !Fn−1 implies that the coefficient of the identity in .Fn−1 modulo the Kauffman bracket skein relations is equal to .[n − 1] A4 !. By using the inductive assumption and Corollary 9.3.5, we obtain the desired result: .

Fn Fn = [n] A4 Fn−1 Fn = [n] A4 [n − 1] A4 !Fn = [n] A4 !Fn . o

By Theorem 9.3.6, we see that .Fn is “almost” an idempotent element in .TLn . By using the same approach as before, we normalize the A-symmetrizer. The resulting symmetrizer modulo the Kauffman bracket skein relations is an idempotent element in .TLn and is known as the Jones-Wenzl idempotent. Definition 9.3.7 Let .Q(A) denote the field of rational functions in the variable A, whose elements are functions of the form .P/Q where .P, Q ∈ Z[A±1 ]. Define a normalized A-symmetrizer (succinctly, an A-symmetrizer) . fn ∈ Q(A)Bn defined by the following formula: 1 Fn, . fn = [n] A4 ! where .[n] A4 = 1 + A4 + A8 + · · · + A4(n−1) =

A4n −1 A4 −1

and .[n] A4 ! =

n | |

[i] A4 .

i=1

Our normalization is chosen so that . fn is an idempotent element in .TLn . The most recognized name for this A-symmetrizer is the Jones-Wenzl idempotent. By definition of . fn we obtain the following two corollaries. Corollary 9.3.8 . fn ei = ei fn = 0.

9.3 The Jones-Wenzl Idempotent

175

Corollary 9.3.9 . fn fn = fn .

Fig. 9.1: Jones-Wenzl idempotent, . fn Diagrammatically, we represent . fn as a square with n strands entering and n strands exiting, as shown in Fig. 9.1. In the diagrams to follow, any strand labeled with n denotes n parallel copies of the strand. Example 9.3.10 Since . f2 =

1 [2] A4 ! F2 ,

then

.

where .Δn is the nth Chebyshev polynomial of the second kind in the variable d (see Lecture 6). Exercise 9.3.11 Compute .F3 and . f3 from the definition. Confirm that .F3 is equal to

.

= [3] A4 ! · 13 + A2 [2]2A4 (e1 + e2 ) + A4 [2] A4 (e1 e2 + e2 e1 ), and .

f3 = 13 +

A2 [2] A4 (e1 + e2 ) + A4 (e1 e2 + e2 e1 ) . [3] A4

Next, we give a few corollaries that are direct results of the theorems and the corollaries to .Fn . The first corollary is a direct result of Corollary 9.3.5, that is, . fn multiplied by any positive braid word, modulo the Kauffman bracket skein relations, is just a scalar multiple of . fn . Corollary 9.3.12 Let .π ∈ Sn , then

176

9 Symmetrizers of Finite Groups and the Jones-Wenzl Idempotent .

fn bπ = λb π fn,

where .λb π ∈ Z[A±1 ] is the coefficient of the identity element in the linear combination of .bπ modulo the Kauffman bracket skein relations . Proof This follows from the definition of . fn and Corollary 9.3.5.

o

The next corollary tells us that . fn is a linear combination of elements of .TLn and the coefficient of the identity element is equal to one. n−1 . Corollary 9.3.13 .( fn − 1) is an element of the algebra generated by .{ei }i=1

Proof By definition, . fn is a linear combination of braids on n strands. By evaluating the braids modulo the Kauffman bracket skein relations, . fn becomes a linear combination of elements of .TLn . By Corollary 9.3.12 and Corollary 9.3.9, the coefficient o of the identity element is equal to one, .λ fn = 1, as desired. Corollary 9.3.14 . fn is an eigenvector under multiplication on any element of .TLn . Let . x ∈ TLn , then . fn x = λ x fn . Proof This follows from the definition of .TLn and Corollary 9.3.8.

.

o

Next is the property of absorption, which states that . fn absorbs any smaller . f j with n − j parallel strands.

Corollary 9.3.15 −

= .

Proof By Corollary 9.3.12, −

= .

where .λ fj is the coefficient of the identity obtained from . f j modulo the Kauffman o bracket skein relations. By Corollary 9.3.13 .λ fj = 1.

9.3 The Jones-Wenzl Idempotent

177

Theorem 9.3.16 ([Wen]) A recursion formula for the nth Jones-Wenzl idempotent, fn , is described in Eq. 9.3:

.

(9.3)

. 2n+2 −A−2n−2

where .Δn = (−1)n A

A2 −A−2

= (−1)n A−2n [n + 1] A4 .

Proof We will first use Corollary 9.3.15 to produce . fn−1 from the last .n − 1 strands coming out of . fn . Next, we use the same argument as in the proof of Theorem 9.3.6, that is, we use fact that any permutation in .Sn can be written as a permutation on the last .n − 1 strands with the first strand placed on the kth position, for some .0 ≤ k ≤ n − 1. Here . k = 0 implies that the first strand does not cross any strands. (Notice for . k = 0, we may use Corollary 9.3.15 again to remove the second copy of . fn−1 .)

.

We may use the Kauffman bracket relation on the first crossing in the last tangle to produce the following,

.

Finally, we may use the Kauffman bracket relation on the k crossings in the last two summations of tangles and then group equivalent tangles. Notice that all B smoothings are annihilated by Theorem 9.3.4 so we only have two distinct tangles from the resulting smoothings.

178

9 Symmetrizers of Finite Groups and the Jones-Wenzl Idempotent

.

After applying the formula, .Δn = (−1)n A−2n [n + 1] A4 , we have the desired result. o

9.4 Properties of Tangles Using the Jones-Wenzl Idempotent The following important basic result follows from Theorem 9.3.16. Corollary 9.4.1 Let .tr1 ( fn ) be obtained from . fn by closing the top string in . fn (see Fig. 9.2). Then Δn .tr 1 ( fn ) = fn−1 . Δn−1

−1

=

−1

Δ Δ −1

Fig. 9.2: Illustration of .tr1 ( fn ) Proof If we close the first string on top of . fn in Eq. 9.3, we have the desired result.

tr1 ( ) = .

−1

−

Δ Δ

−2 −1

−1

−2

.

9.4 Properties of Tangles Using the Jones-Wenzl Idempotent

179

Δn−2 fn−1 Δn−1 Δ1 Δn−1 − Δn−2 = fn−1 Δn−1 Δn = fn−1 . Δn−1

tr1 ( fn ) = Δ1 fn−1 −

.

o Corollary 9.4.2 Let .trAnn ( fn ) be obtained by closing the strings of . fn in the annulus, and let .Sn (z) denote the nth Chebyshev polynomial of the second kind in variable z, where z denotes the nontrivial curve in the annulus. Then trAnn ( fn ) = Sn (z).

.

Proof First we use Theorem 9.3.16 on . fn in the annulus to obtain: −1

=

−1

−

Δ Δ

−2 −1

−2 .

Then we use the idempotent property of . fn to get: −1

=

−2

−

Δ Δ

−2 −1

.

Finally, by Corollary 9.4.1 we have the desired result.

o

Corollary 9.4.3 .trR2 ( fn ) = Δn .1 Proof This is a direct result from Corollary 9.4.2; by closing . fn in .R2 we see that the relation is the nth Chebyshev polynomial of the second kind in the variable d, o which is denoted by .Δn (see Lecture 6). Exercise 9.4.4 Let .bσ1 σ2 σ1 be a minimal positive braid such that . p(bσ1 σ2 σ1 ) = σ1 σ2 σ1 . Show that .trAnn (bσ1 σ2 σ1 ) is proportional to .trAnn ( f3 ). 1 .tr

2

( fn ) is also known as the Markov trace; see [Lic7] for more details.

180

9 Symmetrizers of Finite Groups and the Jones-Wenzl Idempotent

In Lecture 16 we will work with .TLn over .Q(A) where A is a primitive 4rth root of unity. The following Lemma tells us that .trR2 ( fr−1 ) vanishes which will allow us to work with a finite-dimensional vector space. The details of this finite-dimensional vector space will be given in Lecture 16. Lemma 9.4.5 Let A be a primitive 4rth root of unity, then .trR2 ( fr−1 ) = 0 and trR2 ( fi ) = 0 for .0 ≤ i < r − 1.

.

r −1

−2n−2

−1) Proof Since .Δn = (−1) AA2 −A(A , then .Δn = 0 when .n = kr − 1 for . k ∈ Z+ . −2 o Take . k = 1, then by Corollary 9.4.3, we have our desired result. 4(n+1)

A few results involving the Jones-Wenzl idempotent from [Lic8] will be discussed below and are vital preliminary results to Lecture 16. Lemma 9.4.6 ([Lic8])

= (−1)

2 +2

.

Lemma 9.4.7 ([Lic8])

=

,

.

where . ς =

(−1) a ( A2(n+1)(a+1) −A−2(n+1)(a+1) ) . A2(n+1) −A−2(n+1)

Definition 9.4.8 Let .a, b, c be integers and consider a 3-vertex in a diagram with curves labeled .a, b, and c, respectively. The 3-vertex is called admissible if .(a+b−c), (a+b−c) .(b + c − a), and .(a + c − b) are even integers. Let . x = , . y = (b+c−a) , 2 2 and . z = (a+c−b) , then an admissible 3-vertex is defined by using Jones-Wenzl 2 idempotents as shown in Fig. 9.3.

9.4 Properties of Tangles Using the Jones-Wenzl Idempotent

181

:=

Fig. 9.3: Admissible 3-vertex Definition 9.4.9 A theta net evaluation decorated by .(a, b, c) is the 3-vertex graph given in Fig. 9.4, denoted by .θ(a, b, c).

Fig. 9.4: Decorated theta net .θ(a, b, c) We may rewrite the decorated theta net by using Fig. 9.3 to obtain

(9.4)

.

where .a = y + z, .b = x + z, and .c = x + y. Lemma 9.4.10 ([KL, Lic8]) Let .Δn ! denote the product .Δn ! = Δn Δn−1 · · · Δ1 . Then Γ (x, y, z) =

. R2

Δx+y+z !Δx−1 !Δy−1 !Δz−1 ! . Δx+y−1 !Δx+z−1 !Δy+z−1 !

Lemma 9.4.11 (Bubble Popping) Let .δa,d be the Kronecker delta, then

.

(9.5)

Corollary 9.4.12 ([BIMP1]) The formula for the decorated theta net in .R2 can be naturally generalized to the decorated theta net in the annulus. Let .α denote the homotopically nontrivial curve in an annulus, then

182

9 Symmetrizers of Finite Groups and the Jones-Wenzl Idempotent .

θ Ann (a, b, c) =

Sa (α) θ(a, b, c), Δa

where we start from a pair of pants with holes .∂x, ∂y , and .∂z . Then, we cap off .∂z , so that .∂x is homotopic to .∂y as shown in Fig. 9.5.

(a)

0,3

(b)

).

(

Ann (

).

Fig. 9.5: (a) Is a decorated theta net in a pair of pants with holes .∂x, ∂y and .∂z and (b) is a decorated theta net in an annulus obtained by capping off .∂z

Proof Using Lemma 9.4.11, we obtain

=

(

) Δ

=

(

) Δ

.

which is equal to .Sa (α) the annulus.

θ(a, b, c) , where .α is the homotopically nontrivial curve in Δa o

9.5 Exercises Exercise 9.5.1 Calculate F3 in the basis of TL3 . Exercise 9.5.2 Use Lemma 9.4.6 to find a formula of an unknot with writhe equal to one decorated by fn as shown below.

.

Exercise 9.5.3 For Lemma 9.4.11 explain why Eq. 9.5 is zero when a = d and explain why the left-hand side of Eq. 9.5 is a scalar multiple (λ) of fa when a = d. Then prove Lemma 9.4.11 by taking the trace of Eq. 9.6 and solving for λ.

9.5 Exercises

183

.

=

(9.6)

.

Exercise 9.5.4 Prove the equation given in Fig. 9.6.

=

2

+(

2 −2

−2 −2

−

−1

)

Fig. 9.6: An illustration of “sliding over Chebyshev”

Exercise 9.5.5 Use Lemma 9.4.7 to find the formula for the Hopf link with one component decorated by fn as shown below.

.

Exercise 9.5.6 Use Lemma 9.4.7 to find the formula for the Hopf link with one component decorated with m parallel copies and the second decorated by fn as shown below.

.

Exercise 9.5.7 Prove the equations in Figs. 9.7 and 9.8.

=

−1

+

−

Fig. 9.7: Lattice crossing m × 1

−1

184

9 Symmetrizers of Finite Groups and the Jones-Wenzl Idempotent

=

−1

+

−

−1

Fig. 9.8: The mirror image of the lattice crossing m × 1

Exercise 9.5.8 Find the formula for the crossing of Jones-Wenzl idempotents (as in Fig. 9.9).

Fig. 9.9: Crossing of Jones-Wenzl idempotents

Lecture 10 Plucking Polynomial of Rooted Trees and Its Use in Knot Theory

The plucking polynomial is an invariant of rooted trees with connections to knot theory. It was devised as a tool to analyze lattice crossings after taking the quotient by the Kauffman bracket relations. In this lecture we give an introduction to the plucking polynomial and explore its properties and connections to knot theory. The starting point of the formal computation of the plucking polynomial is the notion of the noncommutative plane, that is, the plane where x and y are only q commuting.

10.1 Introduction The first author conceived the notion of the plucking polynomial after finishing work with Mieczyslaw K. Da¸bkowski and Changsong Li in [DaLiPr] concerning skein modules of lattice crossings which will be introduced in Lecture 11. The plucking polynomial’s connections to lattice crossings are explored in [DP3]. A further study of the properties of the plucking polynomial can be found in [CMPWY1, CMPWY2, CMPWY3, Prz29]. In this lecture we will follow the work of the first author in [Prz28] by giving an introduction to the plucking polynomial along with a few interesting properties. Then we will finish with the plucking polynomial with a delay function and its relation to lattice crossings.

10.2 The Plucking Polynomial The plucking polynomial is a polynomial defined recursively from a plane rooted tree by “plucking” the leaves of the tree. In this section we will first define a rooted tree and give examples of special rooted trees that will play a special role in the next © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_10

185

186

10 Plucking Polynomial of Rooted Trees and Its Use in Knot Theory

section. We will also define the plucking polynomial and show a few elementary calculations. Definition 10.2.1 A plane rooted tree is a graph that is connected and does not contain any cycles (acyclic), and one vertex is designated as the root. A leaf is a vertex with only one edge attached to it. See Figs. 10.1 and 10.2 for examples of rooted trees.

−1

2

1

2

• • •

• • •

• • •

• • •

• • •

1

• • •

edges

edges root (a) An illustration of

root .

(b) An illustration of

−1

1

.

Fig. 10.1: Example of rooted trees: (a) .Tb,a is a rooted tree with 2 branches of length b and a, and each branch has one leaf, .v2 and .v1 , respectively. (b) .Tak ,ak−1,...,a1 is a rooted tree with k branches of length .ai for .1 ≤ i ≤ k, and each branch has one leaf

Definition 10.2.2 Let T be a plane rooted tree and . |E(T)| denote the number of edges of T. The plucking polynomial .Q(T, q) (or succinctly .Q(T)) is the unique polynomial obtained from T by the initial condition and recursive formula shown below: 1. If T is the one vertex tree (. |E(T)| = 0), then .Q(T, q) = 1. 2. If . |E(T)| > 0, then Q(T, q) =



.

qr(T,v) Q(T − v, q),

v ∈L(T )

where the sum is taken over all leaves of T, . L(T) is the set of leaves of T that is vertices of degree one different from the root, .r (T, v) is the number of edges of T to the right of the unique path connecting v to the root, and .T − v is a rooted tree obtained by removing the leaf v from T along with the edge directly connected to v. Example 10.2.3 Consider the rooted tree,

10.2 The Plucking Polynomial

187 2

1

= 3

Then the unique path connecting .vi to the root for .i = 1, 2, 3 is shown in Fig. 10.2, yielding .r (T, v1 ) = 0, .r (T, v2 ) = 1, and .r (T, v3 ) = 3.

2

2

1

1

3

3

3

2

1

Fig. 10.2: The red segments highlight the unique path from .vi to the root for .i = 1, 2, 3 Therefore, .Q(T, q) = q0 Q(T − v1 ) + q1 Q(T − v2 ) + q3 Q(T − v3 ), where 2

−

1

=

− 3

2

1 2

−

=

3

1

=

3

Notice that .Q(T − v1 ) = Q(T − v2 ), so we only need to compute .Q(T − v1 ) and Q(T − v3 ).

.

and

188

10 Plucking Polynomial of Rooted Trees and Its Use in Knot Theory

Therefore, .Q(T, q) = (1 + q)(1 + q + q2 ) + q3 (1 + q) = (1 + q)(1 + q + q2 + q3 ). Notice that in this example we calculated the plucking polynomial of a special rooted tree shown in Fig. 10.1a. That is, .Q(T − v1 ) = Q(T1,2 ) = [3]q , where .[n]q = 1+q+· · ·+q n−1 , is called the quantum integer, as described in Definition 9.3.3. Using quantum integers we have .Q(T, q) = [2]q [4]q . A generalization of this observation is given in the next proposition. Definition 10.2.4 The polynomial   [n]q ! n . = i, n − i q [i]q ![n − i]q ! is called a Gaussian polynomial and has the convention: .[0]q ! = 1, . n = 0. and . −1 Exercise 10.2.5 Calculate .Q(T2,4 ) and verify that it is equal to .

n 0 q

=1=

n

n q,

6

4,2 q .

10.2.1 q-Commutative Polynomials and Special Plucking Polynomials The special plucking polynomial, .Q(Tb,a ), of the rooted tree .Tb,a , as shown in Fig. 10.1a, is similar to the binomial coefficient of Newton’s binomial formula. The second special rooted tree, shown in Fig. 10.1b, .Tak ,...,a1 is a generalization of .Tb,a , where we now extend to k branches of various lengths, with each branch containing only one leaf. This special plucking polynomial has similarities to Sir Isaac Newton’s multinomial formula which we will discover in this section. Proposition 10.2.6

 Q(Tb,a ) =

.

 a+b . a, b q

Proof We will prove this by induction on the sum .a + b. First note that .Q(T0,0 ) =   0 . By the inductive hypothesis on .a + b assume that .Q(Tb,a−1 ) = a+b−1 1 = 0,0 a−1,b q q   and .Q(Tb−1,a ) = a+b−1 a,b−1 q . By definition we have the recursive formula .Q(Tb,a ) =     a a+b−1 Q(Tb−1,a ) + q a Q(Tb,a−1 ) which means that .Q(Tb,a ) = a+b−1 a,b−1 q + q a−1,b q . We only need to verify Eq. 10.1.       a+b−1 a+b a+b−1 . = + qa . (10.1) a, b − 1 q a, b q a − 1, b q First notice that .[a + b]q = [a]q + q a [b]q ;

10.2 The Plucking Polynomial

189

[a + b]q = 1 + q + · · · + q a+b−1

.

= (1 + q + · · · + q a−1 ) + q a (1 + q + · · · + q b−1 ) = [a]q + q a [b]q . We can use this property to produce our desired result:     [ab − 1]q ! [a + b]q ! a+b = [a]q + q a [b]q . = a, b q [a]q ![b]q ! [a]q ![b]q ! [a + b − 1]q ! [a + b − 1]q ! + qa [a − 1]q ![b]q [a]q ![b − 1]q !     a+b−1 a a+b−1 +q . = a, b − 1 q a − 1, b q =

 Notice that for .Tb,a , the leaf from branch a is counted as one but the leaf taken from branch b is counted with a weight of .q a . If the weight of each leaf on branch a and branch b were both equal to one, then we would only need to count the number  of different ways to pluck each leaf. The result is the binomial coefficient . a+b a,b of Newton’s binomial formula  n   n n .(x + y) = x i y n−i, i, n − i i=0 where x and y commute. Newton’s binomial formula can be generalized by weakening commutativity, that is, by considering variables . x, y that q-commute; . yx = qxy where q commutes with x and y. The result which likely has been known since the nineteenth century (compare [Mat]) is given in Proposition 10.2.7. From this proposition we see that .Q(Tb,a ) is equal to the generalized binomial coefficient of Newton’s binomial formula.1 Proposition 10.2.7 If . yx = qxy, then (x + y) =

.

n

n   i=0

n i, n − i

 x i y n−i, q

 n  where the generalized binomial coefficient . i,n−i is an element of the quantum q plane (q-commutative polynomials in the variables x and y over the ring .Z[q]). Proof The proof is left as an exercise.



Exercise 10.2.8 Prove Proposition 10.2.7. 1 The first published account of using q-commutativity relation to get the Gaussian polynomial seems to be given in the papers [Pot] in 1950 and [Schüt] in 1953.

 n  . i, n−i q

190

10 Plucking Polynomial of Rooted Trees and Its Use in Knot Theory

Proposition 10.2.9 

a1 + a2 + · · · + ak .Q(Ta k ,a k−1,...,a1 ) = a1, . . . , ak where .

 a1 +a2 +···+ak  a1,...,a k

q

=

 , q

[a1 +a2 +···+ak ]q ! [a1 ]q ![a2 ]q !···[a k ]q ! .

Proof By definition of the plucking polynomial, .Q(T0,0, ··· ,0 ) = Q( ) = 1and we have the following recursive formula: Q(Tak ,ak−1,...,a1 ) = Q(Tak ,...,a2,a1 −1 ) + q a1 Q(Tak ,...,a2 −1,a1 ) + · · · + q a1 +a2 +···+ak−1 Q(Tak −1,...,a2,a1 ).   2 +···+a k can also be defined by the same initial It is enough to show that . a1 +a a1,...,a k q 0  condition and recursive formula. Since . 0,0, ··· ,0 q = 1, then we only need to prove that     a1 + a2 + · · · + ak a1 + a2 + · · · + ak − 1 . = (10.2) a1 − 1, a2, . . . , ak a1, a2, . . . , ak q q   a1 + a2 + · · · + ak − 1 +q a1 a1, a2 − 1, . . . , ak q   a + a 1 2 + · · · + ak − 1 a1 +a2 +···+ak−1 +··· + q . a1, a2, . . . , ak − 1 q .

This can be done using the same technique in the proof of Proposition 10.2.6 by first noting the following equation holds: [a1, a2 + · · · + ak ]q = [a1 ]q + q a1 [a2 ]q + · · · + q a1 +a2 +···+ak−1 [ak ]q,

.

then by using induction on the sum .a1 + a2 + · · · + ak .

(10.3) 

Exercise 10.2.10 Prove Eq. 10.3, then use Eq. 10.3 to prove Eq. 10.2. Exercise 10.2.11 Calculate .T1,2,3 and verify that it is equal to .



6  1,2,3 q .

Exercise 10.2.12 Confirm that .T1,...,1 = [k]q !, where .T1,...,1 has k branches. .Q (Ta k ,a k−1,...,a1 ) is similar to the multinomial coefficient of Newton’s multinomial formula . If the weight of each leaf on every branch were counted as one, then we would again only need to count the number of different ways to pluck each leaf resulting in the multinomial coefficient of Newton’s multinomial formula    a1 + a2 + · · · + ak a1 a2 n x1 x2 · · · xkak , .(x1 + x2 + · · · + xk ) = a1, . . . , ak 

{a1,...,a k ;

ai =n}

10.2 The Plucking Polynomial

191

where . xi commute for .1 ≤ i ≤ k. As before, Newton’s multinomial formula can be generalized by replacing commutativity with q-commutativity, . x j xi = qxi x j for .i < j. The result shown in Proposition 10.2.13 is called the q-multinomial formula. We can relate the q-multinomial coefficients to q-plucking the tree .Tak ,ak−1,...,a1 which yields .Q(Tak ,ak−1,...,a1 ). Proposition 10.2.13 (x1 + x2 + · · · + xk )n =



{a1,...,a k ;

where .



 n a1,...,a k q





.

ai =n}

n a1, . . . , ak

 q

x1a1 x2a2 · · · xkak ,

[n]q ! [a1 ]q ![a2 ]q !···[a k ]q ! .

=

Proof The proof is standard and an inductive proof uses Eq. 10.2.



10.2.2 Properties of the Plucking Polynomial An important result about the structure of the plucking polynomial is the wedge product formula for rooted trees defined by gluing two rooted trees together by their roots. Example 10.2.14 If we let

1

=

and

2

=

, then

1

2

=

Theorem 10.2.15 Let .T1 ∨ T2 be a wedge product of the rooted trees .T1 and .T2 , then   E1 + E2 .Q(T1 ∨ T2 ) = Q(T1 )Q(T2 ), E1, E2 q where .Ei = |E(Ti )|. Proof We will proceed by a double induction on .E1 and .E2 . It is trivially true for E1 = 0 or .E2 = 0. Assume that the formula holds for .E1 and .E2 − 1, and it holds for . E1 − 1 and . E2 . Let .T = T1 ∨ T2 and assume that . E1 and . E2 are not zero, then  .Q (T) = qr(T,v) Q(T − v) .

v ∈L(T )

=



v ∈L(T1 )

qr(T1,v)+E2 Q((T1 − v) ∨ T2 ) +

 v ∈L(T2 )

qr(T2,v) Q(T1 ∨ (T2 − v)).

192

10 Plucking Polynomial of Rooted Trees and Its Use in Knot Theory

By the inductive hypothesis on .E1 and .E2 and reorganizing the terms in the summand     E1 + E2 − 1 E2 E1 + E2 − 1 .Q(T) = q Q(T2 )Q(T1 ) + Q(T1 )Q(T2 ). E1 − 1, E2 q E1, E2 − 1 q 

By using Eq. 10.1, we have our desired result.

Many corollaries follow when we generalize to a finite wedge product of rooted trees. Let .T1, T2, . . . , Tk be plane rooted trees and .T = T1 ∨ T2 ∨ · · · ∨ Tk , then the following corollaries hold. Corollary 10.2.16  Q(T) =

.

Ek + Ek−1 + · · · + E1 Ek , Ek−1, . . . , E1

 Q(Tk )Q(Tk−1 ) · · · Q(T1 ), q

where .Ei = |E(Ti )|. Proof The result follows by applying Theorem 10.2.15 . k − 1 times and noticing that the q-multinomial coefficient is a product of q-binomial coefficients.  The next corollary is a state product formula for .Q(T) utilizing subtrees of T and its decomposition into a wedge product of trees. A subtree of T denoted .T v is the upper portion of T from a vertex of T, denoted by v, where v is the root of .T v . The canonical wedge product decomposition of .T v is the wedge product of . k v trees v .T ∨ · · · ∨ T2v ∨ T1v such that only one edge of .Tiv has as its vertex the root of .T v for kv .1 ≤ i ≤ k v . In fact, all rooted trees have a canonical wedge product decomposition, as shown in Fig. 10.3, which we will succinctly call the wedge product decomposition. Example 10.2.17 Let T be a rooted tree such that

=

Then there is only one nontrivial subtree of T, that is, .T v , shown below in red. The   rest are either T, .T root = T, or a subtree .T v such that . |E(T v )| = 0.

10.2 The Plucking Polynomial

193 −1

2 1

•

•

•

root Fig. 10.3: The canonical wedge product decomposition .T = Tk ∨ Tk−1 ∨ · · · ∨ T2 ∨ T1

=

, therefore

=

The wedge product decomposition of .T v can be written in the form .T2v ∨ T1v , where

1

=

and

2

=

then .Q(T) = 1 + q and

Example 10.2.18 Notice that if

.This observation can be further investigated; it turns out that the plucking polynomial of any rooted tree with only one edge connected to r, say e, is equal to the plucking polynomial of the rooted tree of .T v where v is the vertex connected to e and not equal to r. Exercise 10.2.19 Suppose T is a tree with only one edge, e, extending directly from the root. Let v denote the vertex connected to e and not equal to r; prove that Q(T) = Q(T v ).

.

The next corollary uses Exercise 10.2.19 to convert the plucking polynomial of each rooted tree from the wedge product decomposition of T into the plucking polynomial of a subtree of T. Corollary 10.2.20 .Q(T) can be written as a state product formula  .Q (T) = W(v), v ∈V (T )

194

10 Plucking Polynomial of Rooted Trees and Its Use in Knot Theory

where .V (T) is the set of all vertices of T and .W (v) is the weight of a vertex .v ∈ V(T), called the Boltzmann weight, defined by   E(T v ) .W(v) = . E(Tkvv ), . . . , E(T1v ) q

Proof The corollary follows by taking the wedge product decomposition of T, applying Corollary 10.2.16, then Exercise 10.2.19, and repeating the process on each subtree until you are left with a product of q-multinomial coefficients equal to   .W (v ) for .v ∈ V(T).  The next corollary is important in that it removes the necessity of embedding the rooted tree in a plane. Corollary 10.2.21 The plucking polynomial .Q(T) does not depend on a plane embedding; therefore, it is an invariant of rooted trees. Proof This follows directly from Corollary 10.2.20 since the state product formula does not depend on the embedding. 

=

1

2

1

2

Fig. 10.4: Tree T with two vertices .v1 and .v2 connected to an edge e such that two subtrees of T, .T1 and .T2 , are attached to the vertices .v1 and .v2 , respectively

Corollary 10.2.22 (Change of Root) Let e be an edge of a tree T with the endpoints v1 and .v2 . Let .T1 be a tree from T with root .v1 and .T2 be a tree from T with root .v2 as shown in Fig. 10.4. Denote by .Q(T, vi ) the plucking polynomial of T with root .vi then [|E(T1 )| + 1]q .Q(T, v1 ) = Q(T, v2 ). [|E(T2 )| + 1]q

.

The next set of corollaries relate plucking polynomials to q-binomial coefficients, cyclotomic polynomials, and palindromic polynomials, Corollary 10.2.23 .Q(T) is a product of q-binomial coefficients. Proof This follows directly from Corollary 10.2.20.



10.2 The Plucking Polynomial

195

Corollary 10.2.24 .Q(T) is a product of cyclotomic polynomials.2 Proof This follows directly from Corollary 10.2.20 and since .[n]q = Φd (q). d |n, d1

q n −1 q−1

= 

Corollary 10.2.25 .Q(T) is of the form .c0 + c1 q + · · · + c N q N , where: 1. .c0 = 1 = c N and .ci > 0 for every .i ≤ N. 2. .Q(T) is a palindromic polynomial (also known as ) , .ci = c N −i . 3. The sequence .c0, c1, . . . , c N is unimodal. For . j =  N2 , .

c0 ≤ c1 ≤ · · · ≤ c j ≥ c j+1 ≥ · · · ≥ c N .

4. For a nontrivial tree T (. |E(T)| > 0),  .c1 = (k v − 1), v ∈V (T )

where . k v = degT v (v) is the number of edges attached to v in the subtree .T v . In particular, if T is a binary tree (a rooted tree with the property; for every .v ∈ V(T), .degT v ≤ 2 ), then .c1 is the number of vertices of T which are not leaves. Proof 1. The constant term is obtained in a unique way by always plucking the rightmost leaf from the tree, thus .c0 = 1. The highest power of q is obtained uniquely by always plucking the leftmost leaf of the tree, thus .c N = 1. .ci > 0 for every .i ≤ N follows from a stronger result from the proof of (3). 2. Since q-binomial coefficients are symmetric and a product of s is also symmetric, then the result follows by Corollary 10.2.23. 3. This follows from the observation that a product of symmetric positive unimodal polynomials is symmetric (see [Stan1, Win]) and a result is proven by Sylvester in [Syl] that q-binomial coefficients are unimodal.    a1 +a2 +···+ak  4. Since . a+b a,b q = 1 + q + · · · for .a, b > 0 and . a1,a2,...,a k q = 1 + (k − 1)q + · · · , then the result follows from Corollary 10.2.20. The conditions for .Q(T) to be . strictly unimodal are given in [CMPWY2].

2 The nth cyclotomic polynomial is the minimal polynomial over .Q with root .e2π i/n , denoted We can write this polynomial as .Φ n (q) = (q − ω). An important property to

.Φ n (q).

note is that we can

write .q n

ω n =1, ω k 1 for k < n

− 1 as a product of cyclotomic polynomials, .

d |n

Φ d (q).

196

10 Plucking Polynomial of Rooted Trees and Its Use in Knot Theory

Exercise 10.2.26 Use Corollary 10.2.20 to show that

.

deg(Q(T)) =

 

Eiv E jv  .  v ∈V (T ) 1≤i< j ≤k v 

Exercise 10.2.27 ([CMPWY3]) Let T be a rooted tree with the root r. Prove that Q(T, q) =

.

[E(T)]q ! . v v ∈V (T )\r [E(T )]q

Exercise 10.2.28 Show that:         a1 + a2 + · · · + ak a1 + a2 + a3 a1 + a2 + a3 + a4 a1 + a2 . = ··· a1, a2, . . . , ak a1, a2 q a1 + a2, a3 q a1 + a2 + a3, a4 q q   a1 + a2 + · · · + ak ··· . (10.4) a1 + a2 + · · · + ak−1, ak q

10.3 Kauffman Bracket Motivation for the Plucking Polynomial The plucking polynomial was devised by the first author as a tool to analyze the lattice crossing modulo the Kauffman bracket skein relations; see Fig. 10.5 for .m × n lattice crossing, .Tcr(m×n) .

•••

m

• • •

• • • •••

n Fig. 10.5: Lattice crossing of type .m × n, .Tcr(m×n) Lattice crossings modulo the Kauffman bracket skein relations are a tool to analyze Kauffman bracket skein modules of three-dimensional manifolds (see Lectures 11– 13). They were developed in the series of papers of the first author and Da¸bkowski [DaLiPr, DP3, DP4]. We consider a lattice crossing and resolve every crossing using the Kauffman bracket skein relation relation and then replace each trivial component by .−A2 − A−2 (recall from Lecture 5 that the Reidemeister moves force

10.4 From Catalan Connections to Rooted Trees

197

such a substitution.). As a result we get .Tcr(m×n) as a linear combination of Catalan states (crossingless connection) with coefficients in the ring of Laurent polynomials ±1 . Z[A ]. See Lecture 7 for a description of the Catalan states. A calculation for .Tcr(2×2) is shown in Fig. 10.6.

−4

=

+

2

+

+

−4

+

+

+(

−2

+

2

+

+

−2

+

)

2

−2

+

+

2

+

+

−2

+

+

+

=

−2

+

+

2

−2

+

2

+

+

+

+

2

−2

4

+

+

4

+

Fig. 10.6: Resolving .2 × 2 lattice crossing .Tcr(2×2) by Kauffman bracket skein relations. Notice that the state with coefficient .(A2 + A−2 ) = A−2 [2] A4 is the only state with a non-invertible coefficient Let .Tcr(m×n) =

 C ∈Cat m+n

C(A)C, where the sum is taken over all Catalan states

Cat m+n and the coefficients of each state .C ∈ Cat m+n are a Laurent polynomial C (A) ∈ Z[A±1 ].

. .

Exercise 10.3.1 Find .C (A) for .Tcr(m×2) ; in particular show that .C (A), up to the factor Ai , is a power of .[2] A4 = A4 + 1.

.

10.4 From Catalan Connections to Rooted Trees Drawing a rooted tree from a Catalan state is based on the standard translation going back to the four-color conjecture and its translation from maps to graphs (dual graphs). We adapt it to the Kauffman bracket skein module calculation.

198

10 Plucking Polynomial of Rooted Trees and Its Use in Knot Theory

Definition 10.4.1 For a plane graph G (that is a graph embedded in the plane, .R2 ) we define its dual .G∗ as the plane graph defined as follows: vertices of .G∗ are in bijection with the regions of .R2 − G; we can say informally that they are capitals of connected regions. Edges of .G∗ are in bijection with edges of G and we draw them perpendicularly to edges of G (compare Fig. 10.7). Exercise 10.4.2 Show that regions of .R2 − G∗ are in bijection with vertices of G. Exercise 10.4.3 Our reader probably noticed that our construction of .G∗ is not unique as a plane graph. Show that if we draw G on .S 2 = R2 ∪ {∞} then .G∗ is uniquely embedded in .S 2 . Exercise 10.4.4 Show that the Tait diagram . D(G∗ ) is the mirror image, in xy-plane, ¯ of the diagram . D(G) that is . D(G∗ ) = D(G). Hint: Look at Fig. 10.7.

Fig. 10.7: Dual graphs whose Tait diagrams represent Whitehead links

Exercise 10.4.5 Check that the diagrams of graphs in Fig. 10.7 represent Whitehead links. Use the Jones polynomial to show that Whitehead link is not amphicheiral. Definition 10.4.6 We construct rooted trees from Catalan connections in the following way: (1) Consider a disk D with 2k points on the boundary, say . x1, . . . , x2k , and the Catalan (that is crossingless) connection C. We construct a plane tree .T (C) as follows: arcs of C cut D into . k + 1 regions and we choose vertices of .T (C) to be in bijection with these regions. Two vertices of .T (C) are connected by an edge if the corresponding regions have a common boundary (see Fig. 10.8). (2) If we choose a base point b on the boundary of D (different from . xi ), then the region containing b will correspond to the root of the tree .T(C).

10.4 From Catalan Connections to Rooted Trees

(a) Catalan connection for

= 1.

199

(b) Catalan connections for

(c) Catalan connections for

= 2.

= 3.

Fig. 10.8: Rooted trees from Catalan states with chosen base point b

(3) Consider .∂D divided into two semicircles .∂1 and .∂2 with .b ∈ ∂2 , and let us modify C to .C  by deleting all cords with at least one endpoint on .∂2 . Equivalently we obtained a tree .T (C ) from .T (C) by contracting all edges of .C − C  (this will be the tree considered in calculating .C (A) when C has no returns except the top side, and .∂2 can be identified with the floor of the square tangle). (4) The disk D is now represented as a rectangle with four sides .∂top , .∂ , .∂r , and .∂floor . We first follow the construction from (3) with .∂2 = ∂floor . Then we construct a function from leaves, . L(C ) to .Z+ as follows (we call this function . fL a delay function). If both endpoints of an arc corresponding to a leaf L are in .∂top , then . fL = 1. Otherwise, let .α be the arc corresponding to a leaf, then . fL (α) is the maximal distance of endpoints of .α from .∂top ; compare Fig. 10.9b, c and Fig. 10.10. One of the early applications of the plucking polynomial of a rooted tree without a delay function is given in Exercise 10.5.6. We easily find the formula for the crossing between Jones-Wenzl idempotents using the plucking polynomial; see Exercise 9.5.8. This formula was first discovered by Jim Hoste and the first author in 1992 and independently by Shuji Yamada [Yam1]; compare [DaLiPr]. This plane rooted tree with delay function will be the tree considered in calculating .C (A) when C has no returns on the bottom side.

200

10 Plucking Polynomial of Rooted Trees and Its Use in Knot Theory

Fig. 10.9: Trees .T (C) and .T (C ) from Catalan states

Fig. 10.10: Catalan connections and rooted trees with delayed function on leaves; the bottom tree has delayed function equal to 1, so there is no delay

Definition 10.4.7 The plucking polynomial .Q(T, f ) of a plane rooted tree T with a delay function . f : L(T) → Z+ is given by the initial conditions ,.Q(T, f ) = 0 if . f (w) ≥ 2 for any leaf w, and recursive relation:  .Q(T, f ) = qr(T,v) Q(T − v, fv ), v ∈L1 (T )

where . L1 (T) = f −1 (1) ∈ L(T), and . fv : L(T − v) → Z+ is given by  max{1, f (u) − 1} when u ∈ L(T), . fv (u) = 1 when u  L(T), u ∈ L(T − v).

10.5 Exercises

201

Exercise 10.4.8 Consider the lattice crossing .Tcr(4×4) along with the Catalan state C shown in Fig. 10.9a and its corresponding rooted tree .T (C ) with delay function given in Fig. 10.9c. Let .C (A) be the coefficient of the Catalan state C in .Tcr(4×4) after smoothing all crossings using the initial condition and skein relation from the Kauffman bracket polynomial. Confirm that .C(A) = A12 (1 + A−4 + 2A−8 + A−12 ), and the plucking polynomial of .T (C ) with delay function is .Q(T(C ), f ) = q(1 + q + 2q2 + q3 ). This gives some hint of the relation between the coefficient .C (A) and the plucking polynomial .Q(T, f ) for .q = A−4 ; see [DP3] for full explanation. Remark 10.4.9 Notice that .1 + q + 2q2 + q3 is unimodal but it is not palindromic and not a product of cyclotomic polynomials, and its roots are not roots of unity. Remark 10.4.10 The plucking polynomial of a rooted tree without a delay function is unimodal; see Corollary 10.2.25. However, when we allow a delay function this is not necessarily the case. For example, consider the rooted tree with delay function 1 1 4 44 4 4 1 1 (

)=

Then the plucking polynomial of this rooted tree with delay function f is nonunimodal and equal to Q(T, f ) = q5 + 8q6 + 33q7 + 94q8 + 208q9 + 381q10 + 600q11 + 832q12 +1034q13 + 1171q14 + 1232q15 + 1234q16 + 1212q17 + 1200q18

.

+1212q19 + 1234q20 + 1232q21 + 1171q22 + 1034q23 + 832q24 +600q25 + 381q26 + 208q27 + 94q28 + 33q29 + 8q30 + q31 . = [2]q ![3]q ![6]q !q5 (1 + q7 ). This example was constructed by Alex Landry who at the moment of writing was a student at the James Madison High School in Virginia.

10.5 Exercises Exercise 10.5.1 Verify that Q(T1 ) = Q(T2 ) for the rooted trees T1 and T2 given below:

1

=

and

2

=

202

10 Plucking Polynomial of Rooted Trees and Its Use in Knot Theory

Exercise 10.5.2 Verify that Q(T) = Q(T v ) for rooted tree T and vertex v given below:

=

Exercise 10.5.3 Use the wedge product property given in Theorem 10.2.15 and the formula given in 10.2.13 to compute Q(T) for the rooted tree T given below:

=

Exercise 10.5.4 For the rooted tree T given below, state the Boltzmann weights for each vertex and solve Q(T) as a state product as in Corollary 10.2.20.

=

Exercise 10.5.5 Draw the rooted tree T(C) with base point b and the rooted tree with delay function T(C ) for the Catalan state given below:

=

10.5 Exercises

203

Exercise 10.5.6 Consider the lattice crossing Tcr(m×n) with m ≤ n. Show that the coefficient Ca,b (A) in Tcr(m×n) , where a + b = m and Ca,b is the Catalan state of Fig. 10.11, Ta,b,h is given by the formula:   −(an+(m−a)(2a−n)) m .Ca,b (A) = A . a, b A4

{

{

{

{ Fig. 10.11: Tangle Ta,b,h where a = 3, b = 5, m = a + b = 8, h = 2, n = a + b + h = 10

Lecture 11 Basics of Skein Modules

Skein modules are algebraic objects that generalize the skein theory of link polynomials in .S 3 to arbitrary 3-manifolds. Over time they have evolved into one of the most important objects in knot theory and quantum topology having strong ties with many fields of mathematics and physics. In this lecture we will give the reader a tour of the landscape of skein modules.

11.1 Introduction to Skein Modules Skein modules were introduced by the first author in [Prz4] (independently by Vladimir G. Turaev in [Tur3]) with the goal of building an algebraic topology based on knots. The main object used in the theory is called a skein module which is associated to any arbitrary 3-manifold. There are several skein modules that can be associated to a 3-manifold, each of which captures some information about the manifold based on the knot theory that the manifold supports. What all these modules have in common is that they generalize the skein theory of the various link polynomials in .S 3 , such as the Alexander, Jones, Kauffman bracket, HOMFLYPT, and Kauffman 2-variable polynomial link invariants,1 to any 3-manifold. Thus, the theory of skein modules, as introduced by the first author, can be seen as the natural extension of all the polynomial link invariants in .S 3 to arbitrary 3-manifolds. For a detailed history of the development of skein modules, we refer the reader to [Prz15, Prz31, HP3], and [BPW]. One of the goals of our book is to introduce readers to the rich and deep theory of skein modules and it is our endeavor to spark interest in this field among our readers. We will not include many proofs, but rather we hope to inform the reader of the various directions that research in this area has taken. In this lecture we will survey a few of the many different kinds of skein modules that have been studied in literature. Lectures 12–14 will be devoted in their 1 See Lectures 5 and 6 for a discussion of polynomial link invariants. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_11

205

206

11 Basics of Skein Modules

entirety to the study of the Kauffman bracket skein module, which to date is the most comprehensively studied and best understood skein module. In the most general setting, a skein module is an algebraic object associated to a manifold, usually constructed as a formal linear combination of embedded (or immersed) submanifolds, modulo locally defined relations. In a more restricted setting, the skein module of a three-dimensional manifold is a module composed of linear combinations of links in the manifold, modulo properly chosen relations. While choosing these relations one takes into account the following factors: (i) Whether the module we obtain is computable (ii) How well the skein module distinguishes between different 3-manifolds and different links in the 3-manifolds (iii) Whether the skein module reflects the topology or geometry of 3-manifolds (e.g., the existence of incompressible or non-separating surfaces in a manifold, or geometric decomposition of a manifold) (iv) Whether the skein module admits some additional structure (e.g., filtration, gradation, multiplication, Hopf algebra structure, or categorification) We begin by giving a fairly general definition of a skein module of a 3-manifold M based on oriented tangles2 up to ambient isotopy. However, one soon realizes that there is no end to the amount of generality that can be introduced. At this point in the development of the theory, it seems clear that there is no ultimate definition. There is simply too big of a divide between the modules that can be defined and the modules that are actually feasible to compute and understand for reasonably interesting manifolds. In order to replace .S 3 with an arbitrary 3-manifold M, one must move away from the usual projections and diagrams used to define polynomial link invariants in .S 3 . This can be done by considering links that are identical outside some ball . B3 contained in M and appear as various prescribed tangles inside . B3 . One often wants to distinguish links from one another based on their local appearance in . B3 , and in order to do this it will usually be necessary to assume that M is an oriented manifold. Definition 11.1.1 Let M be an oriented 3-manifold, R a commutative ring with → − unity, . L the set of all oriented links in M considered up to ambient isotopy, and → − → − . R L the free R-module generated by . L. Let .T0, T1, . . . , Tk−1 be a sequence of k oriented tangles in . B3 , and .r0, r1, . . . , rk−1 be a chosen sequence of elements of → − R. Define S to be the submodule of . R L generated by all expressions of the form .r0 LT0 + . . . + rk−1 LTk−1 , where . LT0 , . . . , LTk−1 are k oriented links in M that are identical outside some ball . B3 contained in M but appear as the tangles .T0, . . . , Tk−1 2 Recall that an oriented tangle is a properly embedded oriented 1-manifold in . B3 that has a previously chosen, and henceforth fixed, set of inputs and outputs on .∂B3 . Thus, .∂Ti = Ti ∩ ∂B3 is the same oriented 0-manifold for each i.

11.2 The Signed Skein Module

207

→ − inside of . B3 . The quotient module . R L/S is called a skein module of M, which we → − denote by . S (M; R; r0T0 + . . . + rk−1Tk−1 ; isotopy). → − The notation for the skein module . S (M; R; r0T0 + . . . + rk−1Tk−1 ; isotopy) is often abbreviated by removing a few of the qualifiers to avoid cumbersome notation. The → − arrow in . S is used to indicate that oriented links are used in the definition of the module. Notice that this definition is topological, three-dimensional, and independent of the existence of any particular projection of the 3-manifold to a two-dimensional submanifold. There are numerous ways in which one can alter the definition of a skein module. For example, the most straightforward way, while still retaining a certain degree of generality, is to consider unoriented links or to consider links up to link homotopy. Another important variant is to consider framed links, which are disjoint embeddings of annuli in a 3-manifold. In this case the tangles used in the skein relations must also be framed. In such a situation, it is also usually desirable to introduce additional relations between links that differ only by their framing. Such relations are often called framing relations. The notion of skein modules can be extended to properly embedded 1-manifolds in a 3-manifold with boundary (see [Prz4] and [Prz15]). One can also consider embeddings of surfaces in 3-manifolds and skein relations among them. The Bar-Natan skein module, motivated by BarNatan’s approach to Khovanov homology, is one such example (see [AF, BarN2, Kai3] and Sect. 11.10). For the remainder of the lecture we fix the following notation. Unless otherwise specified, we will always work with an oriented 3-manifold M and a commutative ring R with unity, which may or may not contain some fixed invertible elements. → − The set of ambient isotopy classes of oriented links in M will be denoted by . L, that of unoriented links in M will be denoted by .L, and .L fr will denote the set of → −

unoriented framed links in M up to ambient isotopy. Additionally, .L fr will denote the set of oriented framed links in M up to ambient isotopy.

11.2 The Signed Skein Module → − → − Definition 11.2.1 ([Prz10]) Let S be a submodule of the free R-module R L, generated by the skein expression illustrated in Fig. 11.1. Here L+ and L− denote two oriented links in M that are exactly the same except in a 3-ball in M where they differ as illustrated. The signed skein module is defined as the quotient → − → − → − → − S ± (M; R, L+ − L− ) = R L/ S . It is often abbreviated as S ± (M) for simplicity. Note that since L+ and L− are indistinguishable in the signed skein module, M → − need not be oriented. Thus, S ± (M) is a free R-module over homotopy classes of → − links in M and is, hence, related to the fundamental group of M. If L contains the

208

11 Basics of Skein Modules

+

Fig. 11.1: The signed skein relation → − empty link, denoted by o, then S ± (M) admits a natural algebra structure where the multiplication of two links L1 and L2 is defined as L1 · L2 = L1 u L2 and the empty link serves as the multiplicative identity. This multiplication is associative and commutative and does not depend on the relative position of L1 with respect to → − alg L2 . We denote the signed skein algebra by S ± (M). Note that the ring R embeds in → − alg → − alg S ± (M) by the mapping r |→ r · o. Thus, S ± (M) is isomorphic to the polynomial algebra with coefficients in R and the set of variables, which we denote by π, ˆ is the → − alg set of conjugacy classes of π1 (M). Equivalently, S ± (M) is an R-algebra which is isomorphic to the symmetric tensor algebra over R π, ˆ denoted by SR π. ˆ3

11.2.1 q-Deformation of the Fundamental Group The signed skein module can be quantized (q-deformed) to the class of framed links in a 3-manifold as follows. → −

Definition 11.2.2 ([Prz10]) Let . R = Z[q±1 ] and .S fr be a submodule of the free → −

R-module . RL fr , generated by the skein expressions illustrated in Fig. 11.2. Here, (1) denotes the link obtained by twisting the framing of a link L by a positive full .L twist. The framed version of the signed skein module is defined as the quotient → − fr

→ −

→ −

S± (M; q) = RL fr /S fr . This skein module is also known as the q-deformation of the fundamental group of a 3-manifold.

.

Remark 11.2.3 The ring homomorphism . R −→ Z given by .q |→ 1 induces the → − fr

→ − fr

module homomorphism .S± (M; q) −→ SZπ. ˆ In other words, .S± (M; q) reduces → − → − fr to . S ± (M) when q equals 1, and hence, the name q-deformation for .S± (M; q) is appropriate. 3 The tensor algebra T πˆ is the graded sum

∞ O i=0

T i R π, ˆ where T 0 R πˆ = R, T 1 R πˆ = R π, ˆ and

T i+1 R πˆ = T i R πˆ ⊗ R π. ˆ The symmetric tensor algebra is the quotient SR πˆ = T π/(a ˆ ⊗ b − b ⊗ a).

11.2 The Signed Skein Module

−

209

2

− (1)

−

+

(b) framing relation

(a) skein relation

Fig. 11.2: Relations in the q-deformation of the fundamental group Definition 11.2.4 A surface F embedded in M is said to be non-separating if . M \ F is connected, and separating otherwise. Theorem 11.2.5 ([Prz10]) → − → − fr If M contains no non-separating 2-spheres and tori, then .S± (M; q) = S ± (M; Z)⊗ Z[q±1 ].

The existence of torsion in skein modules is of particular interest as it captures information about the geometry and topology of 3-manifolds. In particular, it captures information about the kinds of surfaces that are embedded in 3-manifolds. The → − fr presence of non-separating tori and 2-spheres in M produces torsion in .S± (M; q) (see Theorem 11.2.6). Theorem 11.2.6 ([Prz10]) → − fr

Let M contain non-separating 2-spheres or tori. Then .S± (M; q) contains torsion elements. In particular: 1. If L is a link in M with the algebraic crossing number . k = 0 with some 2-sphere → − fr

in M, then .(q2k − 1)L = 0 in .S± (M; q). 2. Let . L ' be a link in M with the algebraic crossing number . k = 0 with some nonseparating torus in M. Let L be a link obtained by adding a noncontractible → − fr

curve on the torus to . L '. Then .(q2k − 1)L = 0 in .S± (M; q). → − fr

The link L is not zero in .S± (M; q) since it is homotopically nontrivial in M. In [Kai2], Uwe Kaiser gave a complete description of the framing signed skein → − fr

module. Moreover, he showed that the only method of producing torsion in .S± (M; q) is described in Theorem 11.2.6. Theorem 11.2.7 ([Kai2]) ˆ This set corresponds to Let .b(M) be the set of unordered sequences in .π(M). ˆ Additionally, for each .α ∈ b(M) consider a nonnegative the natural basis for .SR π.

210

11 Basics of Skein Modules

integer .e(α), which is determined by oriented intersection numbers in M. Then we → − R O fr have the following isomorphism of R-modules: .S± (M; q) = . 2e (α)−1 α∈b(M) q The presence of non-separating surfaces in a 3-manifold also leads to torsion in other skein modules, for example, the Kauffman bracket skein module and the framing skein module. These are discussed in more detail in the next section and in Lectures 12 and 15.

11.3 The Framing Skein Module We can consider an even simpler skein module, which only has a framing relation and no skein relation. This skein module is called the framing skein module. Definition 11.3.1 ([HP2]) Let .K fr denote the set of all ambient isotopy classes of unoriented framed knots in M and .S fr be the submodule of the free .Z[q±1 ]-module ±1 ]K fr , generated by the framing expression .K (1) − qK, for any framed knot K .Z[q in .K fr (see Fig. 11.4b). The framing skein module, also known as the first skein module, of M is defined as the quotient: S0 (M; q) = Z[q±1 ]K fr /S fr .

.

In [HP2], Jim Hoste and the first author stated that if M has a non-separating 2-sphere, then .S0 (M, q) is free with basis .K fr . In particular, using the results in [McCu1] they stated the following theorem: Theorem 11.3.2 ([HP2]) For an oriented 3-manifold M, S0 (M; q) = Z[q±1 ]K f ⊕

O

.

K ∈(K fr \K f )

Z[q]

, q2 − 1

where .K f is composed of knots which do not intersect any 2-sphere in M transversely at exactly one point. This theorem was later proved by Vladimir Chernov in [Cher] using a different approach. He used Vassiliev-Goussarov invariants to construct affine self-linking numbers of knots, using which he showed that the number of framings of a knot in an oriented 3-manifold is infinite unless the manifold contains a non-separating 2sphere. The authors of [CCS] gave another proof of this theorem using McCullough’s work in [McCu2]. The authors of this book generalized Theorem 11.3.2 to all links in an oriented 3-manifold [BIMPW]. See Lecture 15 for a discussion.

11.4 The Second Skein Module

211

11.4 The Second Skein Module We now explore skein modules whose relations involve smoothings of crossings. → − → − Definition 11.4.1 ([HP3]) Let .S2 be the submodule of the free R-module . R L generated by the skein expressions illustrated in Fig. 11.3:

+

0

Fig. 11.3: Skein relation for the second skein module Here the pair .(L+, L0 ) denotes two oriented links in M that are exactly the same except in a 3-ball in M, where they differ as illustrated above. We define the second − → → − → − skein module to be the quotient .S2 (M; R) = R L/S2 . The following theorem about the structure of the second skein module was proved by John McCarthy and the first author and formalized in [HP3]. In particular, it was shown that the second skein module is isomorphic to the group ring over the first homology of M with integer coefficients. Theorem 11.4.2 ([HP3])

→ − Let . |L| be the class of L in .H1 (M, Z) and .φ : R L −→ RH1 (M, Z) be the epimorphism of R-modules given by .φ(L) = |L|. Then .φ can be factored through an − → isomorphism .φ- : S2 (M; R) −→ RH1 (M, Z). The proof of this theorem involves proving the following two results: (i) If two oriented links, . L1 and . L2 , represent the same element in .H1 (M, Z), then . L1 u −L2 is the boundary of an oriented surface contained in M. (ii) If . L1 u −L2 is the boundary of an oriented surface in M, then the link . L2 can be reached starting from . L1 via a sequence of elementary operations, which are either modifications of . L+ into . L0 or their inversions. In particular, these results show that any two homologous links in M represent − → the same element in .S2 (M; R). Theorem 11.4.2 was used to prove the following

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11 Basics of Skein Modules

formula by W. B. Raymond Lickorish and Kenneth C. Millett [LiMi] concerning the Kauffman 2-variable polynomial .FL of links in .S 3 : ( ) E −1 c(L) −4lk(S, L\S) . FL (a, −(a + a )) = (−1) a (11.1) /2, S ⊂L

where .c(L) is the number of components of L, the summation is taken over all sublinks S of L (including .S = o), and .l k(S, L \ S) is the linking number between S and . L \ S. This formula was also proved by Turaev in [Tur2]. See Lecture 6 for the definition of the Kauffman 2-variable polynomial. − → Remark 11.4.3 .S2 (M; R) has the structure of an algebra, the multiplicative identity of which is the empty link. The product of . L1, L2 ∈ M is defined as . L1 · L2 = L1 u L2 . − → This multiplication is well defined since . L+ = L− in .S2 (M; R). Moreover, it is associative and commutative and does not depend on the relative position of . L1 with respect to . L2 .

11.4.1 The q-Homology Skein Module We now define the framed version of the second skein module, which is often called a q-deformation of the first homology group. → − fr

Definition 11.4.4 ([Prz12]) Let . R = Z[q±1 ] and .S2 be the submodule of the free → −

R-module . RL fr generated by skein expressions illustrated in Fig. 11.4. The quotient → −

.

→ − fr

→ − fr

RL fr /S2 is called the q-homology skein module and it is denoted by .S2 (M; q).4

-q

-q (1)

0

+

(b) framing relation

(a) skein relation

Fig. 11.4: Relations in the q-homology skein module From the proof of Theorem 11.4.2, we see that any two homologous framed links → − fr

in M differ by only a power of q in .S2 (M; q). 4 We note that one can define the q-homology skein module for any commutative ring with unity and a fixed invertible element q.

11.4 The Second Skein Module

213

Theorem 11.4.5 ([Prz10]) The q-homology skein module is a free .Z[q±1 ]-module if M is a rational homol→ − fr

ogy 3-sphere5 or its compact submanifold. In this case .S2 (M; q) is isomorphic to − → ±1 ]. .S2 (M; Z) ⊗ Z[q The following theorem shows that the presence of non-separating closed surfaces → − fr

in M always yields torsion in .S2 (M; q). Theorem 11.4.6 ([Prz12]) → − fr

S2 (M; q) = Z[q±1 ]T(H1 (M, Z)) ⊕

.

O

Z[q±1 ]/(q2mul(α) − 1).

α∈H1 (M, Z)\T (H1 (M, Z))

T (H1 (M, Z)) and the multiplicity of .α, .mul(α), are defined as follows:

.

Let .φ : H1 (M, Z) × H2 (M, Z) −→ Z be the bilinear form of the intersection of 1cycles with 2-cycles on M. Then .α ∈ T(H1 (M, Z)) if and only if .φα (β) = φ(α, β) = 0 for any . β. The multiplicity of .α, .mul(α), is 0 if .α ∈ T(H1 (M, Z)). Otherwise, .mul(α) is defined as the positive generator of .imφα (H2 (M, Z)). Notice that for a closed 3-manifold M, .T (H1 (M, Z)) is the torsion part of .H1 (M, Z). Corollary 11.4.7 If M is a closed, oriented 3-manifold, then the free part of → − fr

S2 (M; q) is finitely generated.

.

This corollary is of considerable interest as it is an analogue of Witten’s finiteness conjecture for Kauffman bracket skein modules (see Lecture 12). In summary, a non-separating .S 2 in M is detected by torsion in .S0 (M; q), a non-separating .T 2 in M → − fr

is detected by torsion in .S± (M; q), and any non-separating oriented closed surface → − fr

in M is detected by torsion in the skein module .S2 (M; q). More complicated skein modules detect, to varying degrees, separating surfaces as well. See Sect. 12.5 in Lecture 12. We end this section with a discussion of the algebra structure of the q-homology → − fr

skein module. Let . M = F × [0, 1]. We define the product . L1 · L2 in .S2 (M; q) of any two elements . L1, L2 ∈ M by placing . L1 over . L2 in .F × [0, 1] with respect to the height .[0, 1]. The empty link is the identity element of this multiplication.

5 A rational homology 3-sphere is a closed oriented 3-manifold whose rational homology groups are the same as those of the 3-sphere. The Poincaré homology sphere is one such example. See Appendix A for a discussion.

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11 Basics of Skein Modules

Definition 11.4.8 1. A Poisson algebra is a commutative algebra equipped with a Lie bracket, which satisfies the Leibniz rule .[ab, c] = a[b, c] + [a, c]b. 2. Let P be a Poisson algebra over .Z and let A be an algebra over .Z[q±1 ], which is free as a .Z[q±1 ]-module. A .Z-module epimorphism .ψ : A −→ P is called a Drinfeld-Turaev quantization of P if: (a) .ψ(p(q), a) = p(1)ψ(a), for all .a ∈ A and . p(q) ∈ Z[q±1 ]. (b) .ab − ba ∈ (q − 1)ψ −1 ([ψ(a), ψ(b)]), for all .a, b ∈ P. Theorem 11.4.9 ([Prz12]) → − fr

The .Z-algebra epimorphism .ψ : S2 (M; q) −→ ZH1 (M; Z) given by .ψ(L) = |L| is a Drinfeld-Turaev quantization of the group algebra .ZH1 (M; Z) equipped with the Poisson bracket .[α, β] = 2alg(α, β)αβ, where .alg(α, β) is the algebraic crossing number of .α, β ∈ H1 (M; Z). We now turn our attention to the skein module that generalizes the HOMFLYPT polynomial to arbitrary 3-manifolds.

11.5 The HOMFLYPT Skein Module → − → − Definition 11.5.1 Let R = Z[v ±1, z±1 ], L contain the empty link o, and S3 be the → − submodule of the free R-module R L generated by the skein expression illustrated in Fig. 11.5.

−1

−

−

+

0

Fig. 11.5: The HOMFLYPT skein relation The HOMFLYPT skein module, also known as the third skein module or the Jones-Conway skein module, of M is defined as the quotient: − → − → → − → − S3 (M) = S3 (M; Z[v ±1, z±1 ]; v −1 L+ − vL− − zL0 ) = R L/S3 .

.

11.5 The HOMFLYPT Skein Module

215

The existence of the HOMFLYPT polynomial PL (v, z) can be interpreted in the language of skein modules as the following theorem: Theorem 11.5.2 ([FHLMOY, PT1]) − → S3 (S 3 ) = Z[v ±1, z ±1 ] · o, where the empty link o is a generator of the module. In −1 particular, L = PL (v, z)O = PL (v, z)( v z−v )o, where PL (v, z) is the HOMFLYPT polynomial of the link L. In 1987, Mark Kidwell and Jim Hoste [HoKi] computed the HOMFLYPT skein module of the solid torus.6 Turaev computed this skein module independently in [Tur3]. In [Prz5], the first author generalized this result and computed the HOMFLYPT skein module of all oriented surfaces times the unit interval. Theorem 11.5.3 ([Prz5])

− → If F is an oriented surface and I is the unit interval, then S3 (F × I) is a free infinitely generated Z[v ±1, z ±1 ]-module.

Theorem 11.5.4 ([Prz5]) Let F be an oriented surface, πˆ 0 denote the set of conjugacy classes of nontrivial elements of π1 (F), R πˆ 0 be the free Z[v ±1, z ±1 ]-module over πˆ 0 , and SR πˆ 0 be the symmetric tensor algebra over R πˆ 0 . Then, there is an R-module isomorphism i : − → SR πˆ 0 −→ S3 (F × I) such that for w ∈ πˆ 0 , i(w) is represented by a knot K in F × I such that the class of K in πˆ 0 equals w. The proof of Theorem 11.5.4 involves a multistep induction incorporating ideas from [FHLMOY, PT1, HoKi, Tur4], and the method used in the classical proof of the Poincaré-Birkhoff-Witt theorem about the universal enveloping algebras of Lie − → algebras. For an oriented surface F, the HOMFLYPT skein module S3 (F × I) has the structure of an algebra, where the product of two elements L1 and L2 is defined by placing L1 over L2 . As before, the empty link is the identity element of the algebra. −−→ alg We denote the HOMFLYPT skein algebra by S3 (F × I). When F is a planar surface −−→ alg so that F × I is a handlebody, S3 (F × I) has the structure of a Hopf algebra.7 This was first conjectured by Turaev in [Tur5] and ultimately proven for all oriented surfaces by the first author in [Prz6]. Theorem 11.5.5 ([Prz6]) −−→ alg S3 (F × I) is an involuntary Hopf algebra. 6

The first version of [HoKi] from March 1987 served as a motivation for the first author to define skein modules in their full generality a month later. 7 For definitions we refer the reader to [Abe].

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11 Basics of Skein Modules

Moreover, Hugh Morton and Peter Samuelson [MoSa] showed that the HOMFLYPT skein algebras of T 2 × I and F0,2 × I are related to the elliptic Hall al−−→ alg gebra [BS]. Furthermore, S3 (F × I) can be interpreted as a quantization (see −−→ alg [HoKi, Prz4, Tur3, Tur5, Prz6]) and S3 (M) is related to the algebraic set of SL(n, C) representations of the fundamental group of the oriented 3-manifold M (see [Sik3] and [Sik4]). If F is a non-orientable surface, then the twisted I-bundle over F, denoted by F ׈ I, is an oriented 3-manifold and we have the concept of projections of links in F ׈ I onto F. In [Mro1], Maciej Mroczkowski computed the HOMFLYPT skein module of RP2 ׈ I, the twisted I-bundle over RP2 . Theorem 11.5.6 ([Mro1]) → − − → − → S3 (RP2 ׈ I) is freely generated by standard oriented links Ln , where Ln is the link composed of n copies of noncontractible simple closed curves on RP2 as shown in Fig. 11.6. Here L0 is the empty link and RP2 is represented as a disk whose antipodal points are identified.

→ − Fig. 11.6: The standard oriented link L5

Theorems 11.5.4 and 11.5.6 support the following conjecture: Conjecture 11.5.7 If M is a submanifold of a rational homology 3-sphere and does − → not contain a closed, oriented incompressible surface, then S3 (M) is free and isomorphic to SR πˆ 0 . In [GM1], Boštjan Gabrovšek and Maciej Mroczkowski proved this conjecture for the lens spaces L(p, 1).8 Their result is stated in the following theorem. Theorem 11.5.8 ([GM1]) The HOMFLYPT skein module for the lens space L(p, 1) is an infinitely generated free Z[v ±1, z±1 ]-module and it is isomorphic to SR πˆ 0 , where π1 (L(p, 1)) = Z p . − → The problem of computing S3 (L(p, 1)) was first studied by Sofia Lambropoulou and the first author in the currently unpublished paper [LP] (see also [DL, DiLaPr], 8

See Appendix A for a description of lens spaces.

11.6 The k-th Skein Module

217

and [Prz31]), and the results agree with those in [GM1]. We have seen in previous sections that torsion in skein modules is often related to the surfaces embedded in the 3-manifold. The HOMFLYPT skein module is no different in this aspect. In particular, the presence of non-separating 2-spheres in M yields torsion in the HOMFLYPT skein module. Theorem 11.5.9 ([Prz31]) Let S 2 be a non-separating 2-sphere in an oriented 3-manifold M and L be a link (( ) )2 − → v −1 −v 2 in M intersecting S exactly once. Then − 1 L = 0 in S3 (M). z Thus, the HOMFLYPT skein module of any oriented 3-manifold with a nonseparating 2-sphere has torsion. In particular, Patrick Gilmer and Jianyuan Zhong − → [GZ1] computed S3 (S 1 × S 2 ) and showed that it has torsion. They also computed the HOMFLYPT skein module of the connected sum of oriented 3-manifolds in [GZ2] over the field of rational functions. Theorem 11.5.10 ([GZ2]) If M1 and M2 are compact, oriented 3-manifolds and M1 # M2 denotes their connected sum, then over the field of rational functions, − → − → − → S3 (M1 # M2 ) = S3 (M1 ) ⊗ S3 (M2 ).

.

This result was motivated by an analogous result proved by the first author in [Prz18] for the Kauffman bracket skein module of the connected sums of compact oriented 3-manifolds. The proof in [GZ2] follows the same general outline of the proof in [Prz18], which is discussed in Lecture 12. We finish this section with the following conjecture. A similar conjecture for the Kauffman bracket skein module is stated in Lecture 12. Conjecture 11.5.11 Consider the embedding f : B3 c→ M. Then the induced homo− → − → morphism f∗ : S3 (B3 ) −→ S3 (M) is a monomorphism.

11.6 The k-th Skein Module We now define the kth skein module of a 3-manifold for which the first, second, and third skein modules are special cases. → − → − Definition 11.6.1 Let .Sk the submodule of the free R-module . R L generated by linear skein expressions .r0 L0 + r1 L1 + . . . rk−1 Lk−1 , for .r0, r1, . . . rk−1 ∈ R. Here . L0, L1, . . . , Lk−1 are identical links in M except in the parts shown in Fig. 11.7.

218

11 Basics of Skein Modules

0

2

1

1

Fig. 11.7: Link diagrams used in the kth skein module relation − → The .k th skein module of M is defined as the quotient .Sk (M; R)(r0, r1, . . . , rk−1 ) = → − → − R L/Sk . . − → → − Example 11.6.2 .Sk (M; R)(0, . . . , 0) = R L. The kth skein module satisfies the following basic properties. Theorem 11.6.3 ([Prz4]) 1. If . f : M c→ N is an orientation-preserving embedding of 3-manifolds, then f induces a homomorphism .

− → − → f∗ : Sk (M; R)(r0, . . . , rk−1 ) −→ Sk (N; R)(r0, . . . , rk−1 )

of the corresponding skein modules. 2. If . f , g : M c→ N are isotopic embeddings, then . f∗ = g∗ . 3. Let M be a 3-manifold with boundary .∂ M and let .γ be a simple closed curve on 9 . ∂ M. Let N be the 3-manifold obtained from M by adding a 2-handle along .γ and − → . f : M c→ N be the associated embedding. Then . f∗ : Sk (M; R)(r0, . . . , rk−1 ) −→ − → Sk (M; R)(r0, . . . , rk−1 ) is an epimorphism. 4. Let N be the manifold obtained from an oriented 3-manifold M by capping off 2-spheres in .∂ M by 3-cells (in other words, 3-handle addition) and let − → . f : M c→ N be the natural embedding. Then . f∗ : Sk (M; R)(r0, . . . , rk−1 ) −→ − → Sk (N; R)(r0, . . . , rk−1 ) is an isomorphism. A manifold N is obtained from an n-dimensional manifold M by attaching a p-handle .D p ×D n−p to M along its boundary, if . N = M ∪ f D p × D n−p , where . f : ∂D p × D n−p −→ ∂M is an embedding. .D p × {0} is the core of the handle and . {0} × D n−p is the cocore of the handle. See Appendix A for more details. 9

11.7 The Homotopy Skein Module

219

5. If .W = M u N is the disjoint sum of 3-manifolds M and N, then − → − → − → Sk (W; R)(r0, . . . , rk−1 ) = Sk (M; R)(r0, . . . , rk−1 ) ⊗ Sk (N; R)(r0, . . . , rk−1 ).

.

6. (The Universal Coefficient Property) Let R and . R ' be commutative rings with unity, .r0, r1, . . . rk−1 ∈ R, and .ϕ : R −→ → − R ' be a homomorphism. Then the identity map on . L induces the following ' isomorphism of . R (and R) modules: − → − → Sk (M; R)(r0, . . . , rk−1 ) ⊗R R ' = Sk (M; R ')(ϕ(r0 ), . . . , ϕ(rk−1 )).

.

In particular, if . R ' = Z[x0, . . . , xk−1 ], then − → − → Sk (M; R)(r0, . . . , rk−1 ) = Sk (M; R ')(x0, . . . , xk−1 ) ⊗R' R.

.

If .r0, r1, . . . rk−1 are invertible in R, then − → − → ±1 ) ⊗X R, Sk (M; R)(r0, . . . , rk−1 ) = Sk (M; X)(x0±1, . . . , xk−1

.

±1 ]. where . X = Z[x0±1, . . . , xk−1

From Theorem 11.6.3.2 and Example 11.6.2 it follows that links in M are fully − → classified by .Sk (M; Z[x0, x1, . . . , xk−1 ])(x0, x1, . . . , xk−1 ), that is, non-isotopic links represent different elements in the . k th -skein module. This is obviously the case since we allow . x0 = x1 = . . . = xk−1 = 0.

11.7 The Homotopy Skein Module We now define a more tractable invariant of 3-manifolds, which we call the homotopy skein module. This skein module deals with links in M up to link homotopy and not isotopy. Two links are said to be link homotopic in M if one can be deformed into the other by a combination of isotopy as well as by component-wise homotopy. Individual components are allowed to cross through themselves but not each other, that is, we ignore self-crossings in the HOMFLYPT skein module. Definition 11.7.1 Let z be a fixed element in R and H be the submodule of the free → − R-module . R L generated by linear skein expressions . L+ − L− , for a self-crossing, and . L+ − L− − zL0 , for a crossing between different components of . L± . Then the → − → − homotopy skein module is defined as the quotient .H S (M, R; z) = R L/H. The following result is about the homotopy skein module of the product of an oriented surface and the unit interval.

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11 Basics of Skein Modules

Theorem 11.7.2 ([HP2, Tur5]) → − Let F be an oriented surface. Then .H S (F × I, R; z) is a free R-module isomorphic ˆ where .πˆ denotes the conjugacy classes of to the symmetric tensor algebra .SR π, elements of .π1 (F). As in the case of the HOMFLYPT and q-homology skein modules, the homotopy skein module of the product of an oriented surface and the interval has the structure of an algebra, where the multiplication of two elements . L1 and . L2 is defined by placing . L1 over . L2 . The empty link is the identity element of the algebra. We denote −−→ the homotopy skein algebra by .H S alg (F × I). This algebra is related to the universal enveloping algebra of the Goldman-Wolpert Lie algebra of closed curves on surfaces and can be described as a quantization of the symmetric tensor algebra associated to the Goldman-Wolpert Lie algebra. We now describe the Goldman-Wolpert Lie algebra. See [Gol] for details. Let F be an oriented surface, .πˆ denote the set of conjugacy classes of nontrivial ˆ If .α ∈ π, we denote elements of .π1 (F), and . R πˆ be the free R-module with basis .π. its conjugacy class in .πˆ as . |α|. Additionally, if .α is a closed curve in F and p is a simple point on .α, we denote by .αp the loop .α based at p, as well as its homotopy class in .π1 (F, p). Definition 11.7.3 Let .r ∈ R be a fixed element, .sgn(p, α, β) = ±1 denote the oriented intersection number of .α and . β at p as illustrated in Fig. 11.8, and .αp βp denote the product of .α and . β in .π1 (F). Suppose .α and . β are immersed loops in F, which contain only simple points and transverse double points. Define .[α, β] ∈ R πˆ as follows: E .[α, β] = r sgn(p, α, β)|αp βp |. p ∈α∩β

(a) sgn(

)=1

(b) sgn(

) = −1

Fig. 11.8: Oriented intersection number of .α and . β

Theorem 11.7.4 ([Gol]) 1. The operation .[a, b] depends only on the free homotopy classes of .α and . β and, ˆ thus, extends linearly to the bilinear map .[, ] : R πˆ × R πˆ −→ R π. 2. . R πˆ is a Lie algebra under .[, ].

11.7 The Homotopy Skein Module

221

Definition 11.7.5 Let .T (R π) ˆ be the tensor algebra over R associated to the Rˆ The universal enveloping algebra .U R πˆ associated with . R πˆ is defined module . R π. ˆ where .I is the ideal in .T (R π) ˆ generated by elements of the by .U R πˆ = T (R π)/I, ˆ form . x ⊗ y − y ⊗ x − [x, y], for all . x, y ∈ R π. Theorem 11.7.6 ([HP2, Tur5]) −−→ 1. .H S alg (F × I, R; 1) is isomorphic to the universal enveloping algebra .U (R π) ˆ of the Goldman-Wolpert Lie algebra. −−→ 2. .H S alg (F × I, R; z) is a Drinfeld-Turaev quantization of the Goldman-Wolpert Poisson algebra of curves on F.

11.7.1 q-Analogue of the Homotopy Skein Module The q-analogue of the homotopy skein module is of considerable interest. In particular, it distinguishes between certain links with the same HOMFLYPT polynomial. → − Definition 11.7.7 Let .H q be a submodule of the free R-module . R L, generated by the skein expressions . L+ − L− , for a self-crossing, and .q−1 L+ − qL− − zL0 , for a crossing between different components. Then the q-homotopy skein module is −→ → − defined as the quotient .H S q (M, R; q, z) = R L/H q . When . R = Z[q±1, z], we denote the homotopy skein module of M simply as −→ q 3 .H S (M). For . M = S , this skein module is freely generated by trivial links and the value of a link in the module depends only on the number of components and −→ linking numbers between components (see [Prz14]). In particular, .H S q (S 3 ) is freely generated by unlinks .T1, T2, . . ., where .Ti denotes the unlink with i components. Moreover, for an n-component link L in .S 3 , its presentation as . L = w0 (q)Tn + w1 (q)zTn−1 + . . . wn−1 (q)z n−1T1 depends only on the linking numbers between the components of L. An interesting feature of the q-homotopy skein module is that it distinguishes between some links which have the same HOMFLYPT polynomial. Example 11.7.8 In [Birm] it was shown that the 3-component links illustrated in Fig. 11.9 have the same HOMFLYPT polynomial. However, they are different in −→ q 3 2 4 6 2 −1 +q+q 3 +q 5 −q 7 )zT +q 6T .H S (S ). In particular, . L1 = −(1+q +q +q )z T1 +(q 2 3 4 6 8 2 3 5 7 and . L2 = −(q + 2q + q )z T1 + (q + 2q + 2q − q − q9 )zT2 + q6T3 . → − We partially generalize the case when . M = S 3 to . M = F × I. Let .L h = L/(L+ − L− ) denote the set of homotopy links in M and .πˆ denote the set of homotopy ˆ A homotopy link knots in M. Choose some linear ordering .≤ on the elements of .π.

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11 Basics of Skein Modules

1

2

Fig. 11.9: Two 3-component links with the same HOMFLYPT polynomial but with different values in the q-homotopy skein module

L = {K1, K2 . . . . , Kn } in .F × I is said to be a layered homotopy link with respect to the ordering of .πˆ if each .Ki is above .Ki+1 in .F × I and .Ki ≤ Ki+1 . Let .B be the set ˆ including the empty of all layered homotopy links with respect to the ordering of .π, link.

.

Theorem 11.7.9 ([Prz14]) −→ (1) .H S q (F × I) is generated by .B. → − (2) .H S (F × I) is freely generated by .B. −→ (3) If .π1 (F) is abelian, then .H S q (F × I) is freely generated by .B. Furthermore, the q-homotopy skein module has the structure of an algebra when M = F × I, for which the multiplication of two elements is given by stacking, that is, . L1 · L2 is defined by placing . L1 over . L2 in . F × I. The q-homotopy skein module also detects the presence of surfaces with negative Euler characteristic in a 3-manifold. .

Theorem 11.7.10 ([Prz14]) Let F be a surface (not necessary compact) that contains a disk with two holes or a torus with a hole embedded .π1 -injectively, or equivalently, .π1 (F0 ) is not abelian for a connected component .F0 of F (in the compact connected case this means that . χ(F) < 0). Then, −→ (1) .H S q (F × I) has torsion. −→ (2) If .α : RB −→ H S q (F × I) is an R-homomorphism given by .α(L) = L, then .Ker(α) = 0.

11.8 The Kauffman and Dubrovnik Skein Modules

223

Kaiser generalized this theorem and fully characterized oriented 3-manifolds for which the q-homotopy skein module has torsion [Kai1].

11.8 The Kauffman and Dubrovnik Skein Modules We now consider the deformation of the unoriented 2-move, which leads to the Kauffman skein module. It is a generalization of the Kauffman 2-variable polynomial. This skein module must not be confused with the Kauffman bracket skein module which is discussed in detail in Lecture 12. Definition 11.8.1 Let . R = Z[a±1, z±1 ] and .S3,∞ be the submodule of the free Rmodule . RL fr generated by skein expressions in Fig. 11.10. The Kauffman skein

-z

+

0

−

+

-z ∞

(a) skein relation

-a (1)

(b) framing relation

Fig. 11.10: Relations in the Kauffman skein module module is defined as the quotient .S3,∞ (M, R; a, z) = RL fr /S3,∞ .10 The Kauffman skein module has been computed for the following 3-manifolds. Theorem 11.8.2 1. [Kau6] .S3,∞ (S 3 ) = Z[a±1, z±1 ], and the empty link .o is a generator of the −1 module. In particular, . L = FL (a, z)O = FL (a, z)( a+az −z )o. 10 In the notation . S 3,∞ (M, R; a, z), 3 refers to the number of horizontal diagrams in the skein relation, while .∞ refers to the presence of one vertical diagram.

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2. [Tur3] .S3,∞ (S 1 × D2 ) is an infinitely generated R-module. This result was also proved independently by Hoste and Kidwell. 3. [Lie, MPS] If F is an oriented surface with boundary, then .S3,∞ (F × I) is a free R-module. 4. [Mro4] .S3,∞ (L(p, 1)) is an infinitely generated free module when p is odd. When p is even, .S3,∞ (L(p, 1)) contains torsion. In particular, .S3,∞ (RP3 ) contains torsion. Moreover, Zhong computed the Kauffman skein module of the connected sum of compact oriented 3-manifolds. Theorem 11.8.3 ([Zho]) Let . M1 and . M2 be two compact oriented 3-manifolds. Then over the field of rational functions, S3,∞ (M1 # M2 ) = S3,∞ (M1 ) ⊗ S3,∞ (M2 ).

.

We now consider 3-manifolds with framed points11 on their boundary and relative framed links (properly embedded one-dimensional manifolds with framing). This gives rise to the relative version of the Kauffman skein module. Definition 11.8.4 Let M be an oriented 3-manifold and .{x1, x2, . . . , x2n } be a set of 2n framed points on .∂ M. Let .L fr (2n) be the set of all relative framed links in .(M, ∂ M) considered up to ambient isotopy keeping . ∂ M fixed, such that . L ∩ ∂ M = ∂L = {xi }12n . Let .S3,∞ (2n) be the submodule of . RL fr (2n), generated by all the Kauffman skein relations. Then, the relative Kauffman skein module of M is the RL fr (2n) quotient .S3,∞ (M, {xi }12n ) = . S3,∞ (2n) The relative Kauffman skein module for tangles in the disk was computed by Joan Birman, Hans Wenzl, and Jun Murakami and is related to type A Brauer algebra [Brau]. Theorem 11.8.5 1. [Mur, BW, MT] .S3,∞ (D2 × I, {xi }12n ) is a free .(2n−1)(2n−3) · · · 3·1 dimensional R-module. 2. [GoHa] .S3,∞ (F0,2 × I, {xi }12n ) is a free R-module. If we change the signs slightly in the skein relation in Definition 11.8.1, we obtain the Dubrovnik skein module, which generalizes the Dubrovnik polynomial. A framed point in .∂M is an interval in .∂M. A framed arc (ribbon) is a rectangle .arc × I such that end points of .arc × I are the framed points. Thus, a relative framed link intersects .∂M at framed points.

11

11.10 The Bar-Natan Skein Module

225

Dub be the submodule of the free module Definition 11.8.6 Let . R = Z[a±1, z ±1 ] and .S3,∞ fr (1) −aL. Then the . RL generated by the skein expressions . L+ − L− + z(L0 − L∞ ) and . L Dub Dubrovnik skein module, denoted by .S3,∞ (M, R; a, z), is defined as the quotient fr Dub . . RL /S 3,∞

The Kauffman skein module and the Dubrovnik skein module for the solid torus were computed in [Tur3] and are equivalent. Since the Kauffman and Dubrovnik polynomials are equivalent in .S 3 , it is natural to expect them to be equivalent in any arbitrary 3-manifold. In fact, this is true when . M = F × I, for any oriented surface F (see Exercise 11.11.3). However, for lens spaces this was disproved by Mroczkowski in [Mro4]. Theorem 11.8.7 ([Mro4]) Dub (L(p, 1)) is an infinitely generated free module. Compare with Theorem S3,∞ Dub (RP 3 ) = S 3 11.8.2.4. In particular, .S3,∞ 3,∞ (RP ). .

11.9 The (4, ∞)-Skein Module We can also consider skein modules based on skein relations deforming more general tangle moves, for example, rational moves. One such example we discuss is known as the .(4, ∞)-skein module and is based on a deformation of the 3-move. Definition 11.9.1 Let .a, b1, b2, b3, and .b∞ be fixed elements in the ring R such that .a, b0 , and .b3 are invertible. Consider .S4,∞ , the submodule of . RL fr generated by the skein expression .b0 L0 + b1 L1 + b2 L2 + b3 L3 + b∞ L∞ and the framing expression . L (1) − aL. Then the .(4, ∞)-skein module is defined as the quotient fr .S4,∞ (M, R; a, b0, b1, b2, b3, b∞ ) = RL /S4,∞ It had been conjectured that for .S 3 the .(4, ∞)-skein module is generated by trivial links; however, this was disproved in [Prz31] using the concept of Burnside groups of links introduced by Dabkowski and the first author in [DP1]. See Lecture 5 for more details about the Montesinos-Nakanishi 3-move conjecture.

11.10 The Bar-Natan Skein Module The last skein module that we briefly explore in this lecture arises from Bar-Natan’s study of Khovanov homology. An important characteristic of this skein module is that the skein relations involve surfaces embedded in a 3-manifold instead of links.

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Definition 11.10.1 ([AF]) Let M be an oriented 3-manifold, possibly with boundary. A marked surface in M is a properly embedded orientable surface in M decorated with some number of dots. These dots may move between connected components but cannot change components. Let .F (M) be the set of isotopy classes of marked surfaces in M, including the empty surface and let . RF (M) be the free R-module with basis .F (M). Let .S(M) denote the submodule of . RF (M) spanned by the Bar-Natan skein relations described in Definition 11.10.2. The Bar-Natan skein module of M, denoted by . BN(M, R), is defined as the quotient . BN(M, R) = RF (M)/S(M). Definition 11.10.2 (Bar-Natan Skein Relations)

2∗

•

−

= 0.

We note that under these relations The first two relations apply only to spheres that bound balls in M. The third relation, known as the neck-cutting relation, requires the presence of a compressing disk (see Appendix A) in the surface. Any surface with more than one dot is zero under these relations. In fact, a dot represents the element x in the Frobenius algebra Z[x] . 2 and the absence of a dot represents the identity element .1 ∈ Zx[x] 2 . This is x reflective of the connection between Khovanov homology and Frobenius algebras. See Lecture 19 for more details.

11.10 The Bar-Natan Skein Module

227

Theorem 11.10.3 . BN(S 3 ) = Ro, that is, it is the free module of rank 1 over R with basis the empty surface .o. One may also define Bar-Natan skein modules for manifolds with boundary, using properly embedded surfaces. Definition 11.10.4 A pure state is an element of the Bar-Natan skein module that is represented by a single marked surface. We call a marked surface incompressible if the underlying surface is incompressible. Theorem 11.10.5 ([AF]) .

BN(M; R) is spanned by the pure incompressible states.

From Theorem 11.10.5, it follows that if M has no incompressible surfaces, then BN(M, R) is free of rank 1 on the empty surface. To get interesting examples of the Bar-Natan skein module, the 3-manifolds must contain incompressible surfaces. Consider the 3-manifold .S 1 × S 2 . The incompressible surfaces in .S 1 × S 2 are those whose connected components are all isotopic to .{∗} × S 2 .

.

Theorem 11.10.6 ([AF]) Let . z k denote a disjoint union of k parallel spheres in .S 1 × S 2 that are homologically nontrivial. Let .em be a disjoint union of . z m and a dotted nontrivial sphere (i.e., .m + 1 parallel nontrivial spheres with a dot on one of them). Then 1 2 . BN(S × S , R) = R[z] ⊕ Re0 . The result when . M = T 3 is similar. The nonempty connected orientable surfaces in .T 3 are all tori. Up to isotopy they are in one-to-one correspondence with the pairs of triples of integers .±(p, q, r) that are relatively prime and not all zero and we have the following theorem. Theorem 11.10.7 ([AF]) The module . BN(T 3 ) has a basis consisting of the empty surface, parallel copies of the tori parameterized above without a dot, and single copies of the tori above carrying one dot. In [Rus], Heather Russell computed the Bar-Natan skein module of the solid torus with a boundary curve system consisting of 2n copies of the longitude and showed that it is isomorphic to the homology of the .(n, n) Springer variety. Other interesting examples include the Bar-Natan skein modules of Seifert fibered manifolds, which were computed in [AF]. Furthermore, for each Frobenius algebra there is a skein module whose skein relations involve surfaces in an oriented 3-manifold M that bound a curve system in .∂ M. In [Kai3], Kaiser provided a presentation for such skein modules, the generators of which are incompressible surfaces colored by elements of a generating set of the Frobenius algebra and the relations are determined by the geometry of the manifold and relations of the Frobenius algebra.

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11 Basics of Skein Modules

11.11 Exercises

Exercise 11.11.1 Show that the relation q-deformation of the fundamental group.

−

−1

holds in the

Exercise 11.11.2 Prove the formula in Eq. 11.1. Dub (F × I) and S Exercise 11.11.3 Find an isomorphism between S3,∞ 3,∞ (F × I).

Lecture 12 The Kauffman Bracket Skein Module

Kauffman bracket skein modules occupy a special place in the panorama of skein modules, standing out as tall mountains in a lush mathematical landscape. In this lecture we will discuss properties of the Kauffman bracket skein module, focusing on their structure and the significance of the presence of torsion in these modules. Witten’s finiteness conjecture and its generalization about the structure of skein modules play a central role in this discussion. We also provide several examples of 3-manifolds for which the Kauffman bracket skein module has been computed and briefly review the relative version of this module.

12.1 Introduction The skein module based on the Kauffman bracket skein relation is by far the most extensively studied skein module. The Kauffman bracket skein module (KBSM) was introduced as a generalization of the Kauffman bracket polynomial for links in .S 3 to arbitrary 3-manifolds and as an integral part of a movement, initiated by the first author, to create a theory for links similar in spirit to constructions from homological algebra. This movement led to many definitions of other skein modules, some of which have already been covered in Lecture 11. Eventually, this led to a combinatorial construction of the Witten-Reshetikhin-Turaev topological quantum field theory using the language of skein modules (see Lecture 16). Moreover, in many instances, computations of Kauffman bracket skein modules have established connections between the modules and the geometry and topology of 3-manifolds, for example, whether the 3-manifolds contain incompressible surfaces or not. Due to its fundamental role in these constructions and its intricate ties to knot theory, geometric topology, and hyperbolic geometry, in particular, the .SL(2, C) character variety, the Kauffman bracket skein module has become pivotal in the study of quantum topology. In this lecture we discuss properties of the Kauffman bracket © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_12

229

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12 The Kauffman Bracket Skein Module

skein modules of arbitrary 3-manifolds, give examples of manifolds whose KBSMs have been computed, discuss torsion arising in KBSMs, and finally we discuss relative Kauffman bracket skein modules. We note that the theory of skein modules and algebras is a rapidly burgeoning field, with new and exciting results being furnished almost every day, especially for the Kauffman bracket skein module. Hence, it is virtually impossible for us to cover all the directions that research in this area has taken. Relations of the Kauffman bracket skein algebra with the quantum Teichmüller space, the theory of stated skein modules and algebras, the resolution of the positivity conjecture, and more recent work on generalizations of the Kauffman bracket skein module, which are related to quantum invariants coming from non-semisimple categories, will not be discussed in our book. These relations and generalizations, however, are equally enriching and essential in the theory of skein modules and we refer motivated readers to [BW, Le4, Le3, Que], and [CGP] for a discussion of these topics.

12.2 The Kauffman Bracket Skein Module Definition 12.2.1 Let M be an oriented1 3-manifold, L fr the set of unoriented framed links (including the empty link o) in M up to ambient isotopy, R be a commutative ring with unity, and A a fixed invertible element in R. In addition, let RL fr be the sub be the submodule of RL fr generated by all free R-module generated by L fr and S2,∞ (local) skein expressions of the form: (i) L+ − AL0 − A−1 L∞ (ii) L u O + (A2 + A−2 )L, where O denotes the trivial framed knot and the skein triple (L+ , L0 , L∞ ) denotes three framed links in M, which are identical except in a small 3-ball in M where they differ as shown in Fig. 12.1.2 The Kauffman bracket skein module (KBSM) of M is defined as the quotient: sub S2,∞ (M; R, A) = RL fr /S2,∞ .

.

Notice that L (1) = −A3 L in S2,∞ (M; R, A), where L (1) denotes the link obtained from L by twisting the framing of L by a positive full twist (see Fig. 12.2). This is because the first Reidemeister move changes the isotopy type of the framed link and 1 This definition can be generalized to non-orientable manifolds as well, but then the Kauffman bracket skein module will have a lot of torsion. In particular, the trivial knot will be annihilated by A6 − 1. 2 Due to the framing relation illustrated in Fig. 12.2, we can assume that L+ , L0 , and L∞ are all framed links, that is, embedded annuli in M.

12.3 Properties of Kauffman Bracket Skein Modules

0

+

231

∞

Fig. 12.1: Skein triple for the Kauffman bracket skein module changes the Kauffman bracket by a factor of −A±3 (see Lecture 5). This relation is called the framing relation.

+

3

(1)

Fig. 12.2: Framing relation in the Kauffman bracket skein module Thus, an element of S2,∞ (M; R, A) is the equivalence class of some linear combination of (the isotopy classes of) unoriented framed links in M, and it is often referred to as a skein in M. For simplicity we use the notation S2,∞ (M) when R = Z[A±1 ]. See Fig. 12.3 for an example of a skein in S2,∞ (Ann × [0, 1]).

12.3 Properties of Kauffman Bracket Skein Modules In this section we discuss several important properties of Kauffman bracket skein modules including a description of the KBSM of any compact 3-manifold using generators and relations. Theorem 12.3.1 ([Prz4, Prz15, HP4]) 1. Let .i : M c→ N be an orientation-preserving embedding of 3-manifolds. This yields a homomorphism of skein modules .i∗ : S2,∞ (M; R, A) −→ S2,∞ (N; R, A). This correspondence leads to a functor from the category of 3-manifolds and orientation-preserving embeddings (up to ambient isotopy) to the category of R-modules with a specified invertible element . A ∈ R.

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12 The Kauffman Bracket Skein Module

Fig. 12.3: An element of the skein module of the thickened annulus .Ann × [0, 1]; here . x 2 denotes two copies of the homotopically nontrivial curve x in the annulus

2. (i) If N is obtained from M by adding a 3-handle to M (i.e., capping off a hole), and .i : M c→ N is the associated embedding, then .i∗ : S2,∞ (M; R, A) −→ S2,∞ (N; R, A) is an isomorphism. (ii) Let M be a 3-manifold with boundary .∂ M and let .γ be a simple closed curve on .∂ M. Let . N = Mγ be the 3-manifold obtained from M by adding a 2-handle along .γ and .i : M c→ N be the associated embedding. Then .i∗ : S2,∞ (M; R, A) −→ S2,∞ (N; R, A) is an epimorphism. Furthermore, the kernel of .i∗ is generated by the relations yielded by 2-handle slidings. fr More precisely, let .Lgen be a set of framed links in M, which generate .S2,∞ (M; R, A), and . J be the submodule of .S2,∞ (M; R, A) generated by the fr expressions . L − slγ (L), where . L ∈ Lgen and .slγ (L) is obtained from L by sliding it along .γ (i.e., we perform a 2-handle sliding). Then .S2,∞ (N; R, A) = S2,∞ (M; R, A)/J . 3. If . M1 u M2 is the disjoint sum of 3-manifolds . M1 and . M2 , then S2,∞ (M1 u M2 ; R, A) = S2,∞ (M1 ; R, A) ⊗ S2,∞ (M2 ; R, A).

.

4. (The Universal Coefficient Property) Let R and . R ' be commutative rings with unity and .r : R −→ R ' be a homomorphism. Then the identity map on .L fr induces the following isomorphism of . R ' (and R) modules: r : S2,∞ (M; R, A) ⊗R R ' −→ S2,∞ (M; R ', r(A)).

.

12.3 Properties of Kauffman Bracket Skein Modules

233

5. If M is a compact oriented 3-manifold, then M is obtained from a handlebody .Hm of genus m by adding 2- and 3-handles to it (see Appendix A). Thus, the generators of .S2,∞ (M; R, A) are generators of .S2,∞ (Hm ; R, A) and the relators of .S2,∞ (M; R, A) are yielded by 2-handle slidings. In particular, let . M = H1 ∪ F H2 be the Heegaard decomposition of M. Then, .S2,∞ (M; R, A) = S2,∞ (H1 ; R, A) ⊗ S alg (F) S2,∞ (H2 ; R, A).3 Proof (1) .i∗ is well defined because if the framed links . L1 and . L2 are ambient isotopic in M, then .i (L1 ) and .i (L2 ) are ambient isotopic in N. Furthermore, any skein triple .(L+ , . L0 , . L∞ ) in M is sent to a skein triple in N by i. Finally, .i (O) is a trivial framed knot in N. Notice that if .i : M c→ N is an orientation reversing embedding, then .i∗ is a .Z-homomorphism and .i∗ (Aw) = A−1i∗ (w). (2) (i) This result holds because the cocore of a 3-handle is 0-dimensional. (ii) This result holds because the cocore of a 2-handle is 1-dimensional. Also, any skein relation holds in M and the only difference between the KBSMs of M and N lies in the fact that some nonequivalent links in M can be equivalent in N due to the possibility of sliding a link in M along the added 2-handle (i.e., L can be moved from one side of the cocore of the 2-handle to the other side). (3) This is a consequence of the following well-known property of short exact sequences: If .0 −→ A' −→ A −→ A'' −→ 0 and .0 −→ B ' −→ B −→ B '' −→ 0 are short exact sequences of R-modules, then .0 −→ (A' ⊗ B + A ⊗ B ') −→ A ⊗ B −→ fr fr A'' ⊗ B '' −→ 0 is also a short exact sequence. Let .L M and .L N denote the sets of ambient isotopy classes of unoriented framed links in M and N, respectively. fr sub , and . A'' is .S In our case A is . RL M , . A' is .S2,∞ 2,∞ (M; R, A) (similarly for B). Notice that . A ⊗ B corresponds to . RL M ⊗ RL N and . A' ⊗ B + A ⊗ B ' corresponds sub (M u N). to the submodule of skein relations .S2,∞ fr

fr

sub (R, A) −→ RL fr −→ S (4) The exact sequence of R-modules, .S2,∞ 2,∞ (M; R, A) −→ 0, leads to the following exact sequence of . R '-modules (see Proposition 4.5 in [CE]): .

sub S2,∞ (R, A) ⊗R R ' −→ RL fr ⊗R R ' −→ S2,∞ (M; R, A) ⊗R R ' −→ 0.

3 Here, . S alg (F) denotes the Kauffman bracket skein algebra of the surface F times the unit interval. See Lecture 13. See Appendix A for details about Heegaard decomposition.

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12 The Kauffman Bracket Skein Module

Now, by applying the five lemma to the following the commutative diagram with exact rows (see, e.g., Proposition 1.1 in [CE]) sub (R, A) ⊗ R ' −→ RL fr ⊗ R ' −→ S ' S2,∞ R R 2,∞ (M; R, A) ⊗ R R −→ 0 −→ 0 ⏐ ⏐ ⏐ epi ⏐ iso ⏐r . sub ' ' fr ' S2,∞ (R , r(A)) −→ RL −→ S2,∞ (M; R , r(A)) −→ 0 −→ 0

we conclude that .r is an isomorphism of . R '- (and R-) modules. (5) The result follows from parts 2(i) and 2(ii). The KBSM of a handlebody has been discussed in Example 12.4.2 and Theorem 13.2.1. This result implies that computing the KBSM of a compact oriented 3-manifold essentially boils down to finding the exact set of handle sliding relations that generate the submodule . J of .S2,∞ (Hm ; R, A). The following result reduces the problem of finding the generators for the Kauffman bracket skein module, over the field of rational functions, of arbitrary oriented 3-manifolds to prime 3-manifolds. Theorem 12.3.2 ([Prz18]) Let M and N be compact, oriented 3-manifolds, . M # N denote their connected sum, and . Ak − 1 be invertible in R for any . k > 0. Then, S2,∞ (M # N; R, A) = S2,∞ (M; R, A) ⊗ S2,∞ (N; R, A).

.

In particular, the result holds when . R = Q(A), the field of rational functions in the variable A. Proof We give an outline of the proof of this theorem. Any compact oriented 3manifold can be obtained from a handlebody by adding 2- and 3-handles to it. The KBSM of a handlebody is a well-understood free R-module (see Example 12.4.2). Moreover, adding a 3-handle does not change the module and adding a 2-handle adds new relations to the skein module, which are obtained by sliding links along the 2handle (see Theorem 12.3.1.2). Thus, we first consider M and N to be handlebodies . Hm and . Hn , respectively. One can prove the following result. Lemma 12.3.3 ([Prz18]) If D is a meridian disk of a handlebody .Hn , .γ = ∂D, and . Ak − 1 is invertible in R for any . k > 0, then the embedding . j : Hn \ D c→ (Hn )γ induces the isomorphism . j∗ : S2,∞ (Hn \ D; R, A) −→ S2,∞ ((Hn )γ ; R, A) of skein modules. Now, .Hm # Hn is homeomorphic to .(Hm+n )γ , where a 2-handle is added along the boundary of the meridian disk D that separates .Hm from .Hn . Furthermore, . Hm+n \ D = Hm u Hn . Thus, from Lemma 12.3.3 we get that the embedding

12.3 Properties of Kauffman Bracket Skein Modules

235

i : Hm u Hn c→ Hm # Hn yields the isomorphism .i∗ : S2,∞ (Hm u Hn ; R, A) −→ S2,∞ (Hm # Hn ; R, A) of skein modules. The sliding relations are generated by the slidings illustrated in Fig. 12.4. Furthermore, from Theorem 12.3.1.3 we get that .i∗ : S2,∞ (Hm ; R, A) ⊗ S2,∞ (Hn ; R, A) −→ S2,∞ (Hm # Hn ; R, A) is an isomorphism. This proves the result for handlebodies. Note that there is usually an infinite collection of sliding relations. However, when . Ak − 1 is invertible in R and a 2-handle is added along a meridian curve creating a separating .S 2 , one can prove that the relations form a “controllable sequence”, which allows us to reduce all curves cutting the sphere and there are no more relations to be dealt with (see [Prz18]). .

Fig. 12.4: Slidings in . M # N

We can then generalize Lemma 12.3.3 to any compact, oriented 3-manifold. Lemma 12.3.4 ([Prz18]) If D is a properly embedded disk of M, .γ = ∂D, and . Ak − 1 is invertible in R for any . k > 0, then the embedding . j : M \ D c→ Mγ induces the isomorphism . j∗ : S2,∞ (M \ D; R, A) −→ S2,∞ (Mγ ; R, A) of skein modules. We then use Lemma 12.3.4 to show that if .S 2 is embedded in M, then the embedding 2 2 . j : M \S c→ Mγ yields the isomorphism . j∗ : S2,∞ (M \S ; R, A) −→ S2,∞ (Mγ ; R, A) 2 3 of skein modules. Note that .(M # N) \ S and .(M # D ) u (N # D3 ) differ only by parts of their boundaries, so their KBSMs are the same. This immediately gives us that the embedding . j : (M # D3 ) u (N # D3 ) c→ M # N yields an isomorphism 3 3 . j∗ : S2,∞ ((M # D ) u (N # D ); R, A) −→ S2,∞ (M # N; R, A) of the corresponding KBSMs. Another application of Theorem 12.3.1.3 and the fact that KBSMs of M o and . M # D3 coincide give us the desired result. We note that Theorem 12.3.2 does not hold when . R = Z[A±1 ]. See Example 12.4.11. We finish this section with an important result by John W. Barrett in which he proved that the existence of a spin structure4 for any oriented 3-manifold M implies that for any knot K in M we can define its framing, .Spin(K) ∈ Z2 , in such a way that the following theorem holds. 4 For a discussion about spin structure on 3-manifolds, refer to Lecture 15.

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12 The Kauffman Bracket Skein Module

Theorem 12.3.5 ([Bar]) For E any ring R, the map . φ : S2,∞ (M; R, A) −→ S2,∞ (M; R, −A) given by . φ([L]) = (−1) Spin(K) [L], where the sum is taken over all connected components of L, is an R-module isomorphism. In Lecture 15, we show using spin structures that the framing of a link in an oriented 3-manifold cannot be changed by an odd number of full twists on it. We finish this section with the following conjecture. A similar conjecture for the HOMFLYPT skein module is stated in Lecture 11. Conjecture 12.3.6 (Folk Conjecture) Let .i : B3 c→ M 3 be an orientation-preserving embedding of a 3-ball into an oriented manifold . M 3 . Then the map induced on the Kauffman bracket skein modules, .i∗ : S2,∞ (D3 ; R, A) −→ S2,∞ (M 3 ; R, A), is a monomorphism.

12.4 Examples of Kauffman Bracket Skein Modules of 3-Manifolds Computations of skein modules are generally quite difficult and there are few explicit examples of 3-manifolds for which the structure of their Kauffman bracket skein modules is completely known. Often, only the free parts of the skein modules are known. In many cases the KBSMs have been computed only when . R = Q(A). In fact, Witten had conjectured 5 that the Kauffman bracket skein module for any closed oriented 3-manifold over .Q(A) is always finite dimensional. We note, however, that Doug Bullock was the first to observe and remark upon this phenomenon for the KBSM in 1997. In his paper [Bul4], he noted that all known computations at that time showed that the KBSM was finitely generated if and only if the manifold contained no essential surface. Theorem 12.3.2 shows that Witten’s conjecture is stable under connected sums, since if .S2,∞ (M; Q(A)) and .S2,∞ (N; Q(A)) are finite dimensional, then so is .S2,∞ (M # N; Q(A)). In this section we list a few manifolds for which the exact structure of the Kauffman bracket skein module has been computed. Example 12.4.1 ([Kau3]) .S2,∞ (S 3 ) = Z[A±1 ]o. More precisely, .o is the basis element of the module and . L = [L]o = (−A2 − A−2 )< L >o, where .[L] is the unreduced Kauffman bracket polynomial of a framed link L. Moreover, the KBSM of .S 3 with finitely many punctures or balls removed is also .Z[A±1 ]o. In particular, we have that 2 3 ±1 .S2,∞ (S × I) = S2,∞ (R ) = Z[A ]o. 5 The first author heard about Witten’s conjecture from Julien Marché when he visited Oberwolfach in June 2015. Marché had heard about the conjecture from Greg Kuperberg who in turn was asked this question by Edward Witten. In fact, Kuperberg mentioned that Witten had a physical argument for the conjecture (compare with [GiMa]).

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Example 12.4.2 ([Prz4, Prz15, HP4]) .S2,∞ (F × [0, 1]) is a free module generated by the empty link .o and links in F which have no trivial components. Here F is an oriented surface and each link in F is equipped with an arbitrary, but specific framing. This result applies in particular to a handlebody, because .Hn = F0,n+1 × I, where .Hn is a handlebody of genus n and .F0,n+1 is a disk with n holes. The proof of the result for .S2,∞ (F × [0, 1]) is essentially the same as that for the existence of the Kauffman bracket polynomial in .R3 = R2 ×R. A similar result also holds for oriented 3-manifolds that are twisted I-bundles over non-orientable surfaces (.F ׈ I). As a corollary we get that .S2,∞ (RP3 ) = S2,∞ (RP2 ׈ I) is free on two generators: a trivial curve and an orientation reversing curve in .RP2 . Note that .RP3 \ int(B3 ) = RP2 ׈ I. Since .RP3 is homeomorphic to the lens space . L(2, 1), this result is a special case of the following result about the KBSM of lens spaces.6 The next lecture is devoted to discussing the skein modules of surface I-bundles and their corresponding skein algebras. Example 12.4.3 ([HP4]) For . p ≥ 1, .S2,∞ (L(p, q)) is a free .Z[A±1 ]-module and it has . Lp/2] + 1 free generators. To see this, we think of . L(p, q) as obtained from a solid torus .S 1 × D2 by first attaching a 2-handle along the .(p, q)-curve and then a 3-handle. Thus, the natural inclusion .i : S 1 × D2 c→ L(p, q) induces a module epimorphism 1 2 .i∗ : S2,∞ (S × D ) −→ S2,∞ (L(p, q)), given by .i∗ [L] = [i(L)] for a framed link L 1 2 in .S × D . It follows that any generating set for the KBSM of the solid torus, say n ∞ , is a generating set for .S . {i∗ [x ]} 2,∞ (L(p, q)). Here x represents the homotopically n=0 1 2 nontrivial curve in .S × D . However, .i∗ is not a monomorphism (see Theorem 12.3.1.1). In particular, in . L(p, q), by sliding framed links across the 2-handle, one ∞ . Thus, finds many relations between the elements in the generating set .{i∗ [x n ]}n=0 .S2,∞ (L(p, q)) is generated by a fewer number of elements and, in fact, .S2,∞ (L(p, q)) Lp/2] is a finitely generated free .Z[A±1 ]-module with basis .{i∗ [x n ]}n=0 . See [HP4] for details. Example 12.4.4 ([HP5]) .S2,∞ (S 1 × S 2 ) is an infinitely generated .Z[A±1 ]-module. More precisely, ∞ O Z[A±1 ] 1 2 ±1 .S2,∞ (S × S ) = Z[A ] ⊕ . 1 − A2i+4 i=1 This result completes the computation of .S2,∞ (L(p, q)) when . p = 0, since .S 1 × S 2 = L(0, 1). Notice that, .S2,∞ (S 1 × S 2 ) decomposes into the sum of free and torsion parts. Furthermore, when . R = Q(A), we have .S2,∞ (S 1 × S 2 ; R, A) = Q(A). Example 12.4.5 ([HP6]) Let W denote the classical Whitehead manifold,7 which is an open 3-manifold that is contractible but not homeomorphic to .R3 . Then .S2,∞ (W) is infinitely generated, torsion-free, but not free. Moreover, Hoste and the first author 6 See Appendix A for a description of lens spaces. 7 See Appendix A for a description of the Whitehead manifold.

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proved this result for a large class of genus one Whitehead manifolds first discovered by D. R. McMillan in [McM]. Note that this is in stark contrast to the case when 3 . M = R whose KBSM is free on one generator. In fact, Hoste and the first author conjectured the following. Conjecture 12.4.6 ([HP6]) The KBSM of every open contractible 3-manifold other than .R3 is infinitely generated, torsion-free, but not free. Example 12.4.7 ([GH]) Consider the free action of the quaternion group .Q8 on .S 3 . The orbit space of this action is called the quaternionic manifold. It can also be obtained by identifying opposite faces of a cube with one-quarter twists. The KBSM of the quaternionic manifold is a free R-module with five basis elements, where R is the ring .Z[A±1 ] localized by inverting all the cyclotomic polynomials. This result was generalized by Maciej Mroczkowski in [Mro2] to a family of prism manifolds over an arbitrary ring (see Example 12.4.10). Example 12.4.8 Let .Fg,b denote an oriented genus g surface with b boundary components. Thus, .F0,3 denotes the pair or pants and in [DM] it was shown that 1 ±1 .S2,∞ (F0,3 × S ) is an infinitely generated free .Z[A ]-module. Example 12.4.9 In [Car], Alessio Carrega showed that .S2,∞ (T 3 ; Q(A)) is a finitely generated .Q(A)-module with nine generators. Moreover, in [Gil] Patrick M. Gilmer showed that these generators are linearly independent. A more general result was computed in [DW] in which Renaud Detcherry and Maxime Wolff showed that for any closed oriented surface F of genus .g ≥ 2, .S2,∞ (F × S 1 ; Q(A)) is a finitedimensional .Q(A)-module with dimension .22g+1 + 2g − 1. It was earlier shown in [GiMa] that the dimension of this skein module is at least .22g+1 + 2g − 1. Example 12.4.10 Consider oriented 3-manifolds obtained from a twisted I-bundle over the Klein bottle by gluing a solid torus to its torus boundary. This family includes prism manifolds and the quaternionic manifold . We consider a family . Mp of these manifolds whose fundamental group .π1 (MP ) = < x, y | xyx −1 = y −1, x 2 = y p >, that is, .H1 (Mp ) = Z2 ⊕ Z2 if p is even and .Z4 if p is odd. For instance, . M1 is the lens space . L(4, 1), and . M2 is the quaternionic manifold. In [Mro2] it was shown that ±1 .S2,∞ (Mp ) is a free .Z[A ]-module that has .3 + Lp/2] free generators if p is odd and .4 + p/2 free generators if p is even. In particular, for . M1 = L(4, 1), one gets three generators, and for the quaternionic manifold . M2 , one gets five generators (compare with Examples 12.4.3 and 12.4.7). Moreover, for the quaternionic manifold . M2 , this result proves a conjecture in [GH], which states that .S2,∞ (M2 ) is torsion-free. Example 12.4.11 ([BLP]) Let .Hn = F0,n+1 × I denote a genus n handlebody. Then, S2,∞ (H1 # H1 ) = S2,∞ (H2 )/I, where .I is the submodule generated by the expressions . zk − A6 u(zk ), for any even . k ≥ 2. Here . zk ∈ Bk (F0,3 ), where . Bk (F0,3 ) is a subset

.

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239

of a basis of .S2,∞ (F0,3 × I) composed of links that have geometric intersection number k with a disk D that separates the two .H1 s. Furthermore, .u(zk ) is a modification of . zk in the neighborhood of D, as shown in Fig. 12.5. The relation . zk = A6 u(zk ) is a result of the sliding relation . zk = sl∂D (zk ) illustrated in Fig. 12.5. More precisely, the natural epimorphism .i∗ : S2,∞ (H2 )/I −→ S2,∞ (H1 # H1 ), due to the embedding .i : H2 c→ H1 # H1 , is an isomorphism. See Appendix A.

Fig. 12.5: Illustration of the handle sliding relations that generate .I. Here, 6 . sl∂D (zk ) = A u(zk )

Example 12.4.12 The KBSMs of knot and link complements have been computed only for certain classes of knots and links. Thang T. Q. Lê proved that the KBSM over the ring .C[A±1 ] of the complement of a 2-bridge knot .K p is a free module with m

i j basis .{x i y j }, 0 ≤ i, 0 ≤ j ≤ p−1 2 , where . x y denotes the skein represented by the link composed of i parallel copies of the meridian curve x and j parallel copies of the curve y (see [Le1] and Fig. 12.6). One should compare this result with that for .S2,∞ (L(p, q)). The two results are related to each other because . L(p, m) is the double branched cover of .S 3 , branched along the 2-bridge knot .K p . Furthermore, in [LT] m it was shown that the KBSM of the complement of any 2-bridge link is free over the ring .C[A±1 ].

Special cases of the KBSMs of 2-bridge knots had been computed earlier. In [Bul1] Bullock proved this for the KBSMs over .Z[A±1 ] of the complements of torus knots of type .(2, 2p + 1). In addition, Bullock and Walter Lo Faro computed the KBSMs of the knot exteriors of twist knots in [BLF] and found that they were infinitely generated free modules, the bases of which consist of cables of a twocomponent link, one component of which is a meridian of the knot. Example 12.4.13 In [Bul4], Bullock showed that the KBSM of a 3-manifold which is obtained from an integral surgery on the trefoil knot is a finitely generated .Z[A±1 ]module if and only if the manifolds contain no essential surface. Only two manifolds obtained via surgery on the trefoil contain essential surfaces and for these Bullock

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Fig. 12.6: Basis for the KBSM of the complement of 2-bridge knots

showed that the KBSM is infinitely generated. In [Har], John M. Harris pursued a similar vein of research and showed that the KBSMs of some Dehn fillings of ±1 .(2, 2n)-torus links are finitely generated .Z[A ]-modules. Example 12.4.14 ([Det]) Let .K ⊂ S 3 be a knot, .V (K) be its regular neighborhood in S 3 , .S 3 \ K be its knot complement, . μ ⊂ ∂(S 3 \ int(V(K))) be the meridian of K, and 3 . MK (r) be a surgery on K of slope r. If .S2,∞ (S \ int(V(K)), Q(A)[μ]) is finitely generated, then for all except possibly finitely many .r ∈ Q ∪ {∞}, .S2,∞ (MK (r), Q(A)) is finitely generated. Note that with the exception of .(2, 2n + 1)-torus knots, all 2-bridge knots are hyperbolic. Moreover, for a hyperbolic knot, by Thurston’s hyperbolic Dehn surgery theorem all except at most finitely many Dehn fillings are hyperbolic.8 Thus, this example gives infinite families of hyperbolic 3-manifolds whose KBSMs are finitely generated. .

Witten’s conjecture is true for all of the examples listed above. Using factorization algebras, the representation theory of quantum groups, and deformation quantization modules, Sam Gunningham, David Jordan, and Pavel Safronov [GJS] resolved this conjecture in the affirmative. Theorem 12.4.15 ([GJS]) The Kauffman bracket skein module of any closed oriented 3-manifold over the field .Q(A) is finite dimensional. However, it must be noted that when . R = Z[A±1 ], .S2,∞ (M) is not always finite dimensional (see Example 12.4.4). In fact, much less is known about the structure of the Kauffman bracket skein module of an oriented 3-manifold when . R = Z[A±1 ] than when . R = Q(A). Marché had proposed the following conjecture about the structure of the KBSM over . R = Z[A±1 ] (see [DW]). Conjecture 12.4.16 Let M be a closed compact oriented 3-manifold. Then there exists an integer .d ≥ 0 and finitely generated .Z[A±1 ]-modules . Nk so that 8 Refer to Appendix B for a description of Dehn fillings and Dehn surgery.

12.5 Torsion in Kauffman Bracket Skein Modules

241

S2,∞ (M) = (Z[A±1 ])d ⊕

O

.

Nk ,

k ≥1

where . Nk is a .(Ak − A−k )-torsion module for each k. Evidence of the veracity of this conjecture comes from the KBSM of .S 1 × S 2 which splits into the sum of free and torsion parts. Moreover, the proof in [DW] for the KBSM of .F × S 1 implies that the torsion elements are always of . Ak − A−k type, for . k ≥ 1. However, the second author found a counterexample to this conjecture, which is given by the KBSM of the connected sum of real projective spaces. Theorem 12.4.17 ([Bak]) Conjecture 12.4.16 is not true for .S2,∞ (RP3 # RP3 ). In particular, .S2,∞ (RP3 # RP3 ) does not split into the direct sum of free modules and torsion modules. An essential ingredient used in calculating the KBSM of .RP3 # RP3 is the concept of depicting links in .RP3 # RP3 by arrow diagrams. We refer to [DM, GM2, Mro3], and [Bak] for details about the construction of these diagrams. The following theorem gives the exact structure of .S2,∞ (RP3 # RP3 ). Theorem 12.4.18 ([Mro3]) and .t = −A−3 x, that is, t represents x with

Let x denote the arrow diagram .

a negative full twist. Then, .S2,∞ (RP3 # RP3 ) = Z[A±1 ] ⊕ Z[A±1 ] ⊕ Z[A±1 ][t]/S, where S is a submodule of .Z[A±1 ][t] generated by the following two relations: (i) .(An+1 + A−(n+1) )(Sn (t) − 1) − 2(A + A−1 ) (ii) .(An+1 + A−(n+1) )(Sn (t) − t) − 2t

(n−1)/2 E k=1

n/2 E k=1

An+2−4k , for .n ≥ 2 even.

An+1−4k , for .n ≥ 3 odd.

Here, .Sn (t) denotes the Chebyshev polynomial of the second kind. Remark 12.4.19 .S2,∞ (RP3 # RP3 ) does not split as a sum of cyclic modules. This is a direct contradiction to Conjecture 12.4.16, which states that the KBSM of a closed compact oriented 3-manifold over .Z[A±1 ] splits as the direct sums of free modules and cyclic modules. Note, however, that .S2,∞ (RP3 # RP3 ) does contain torsion elements. See [Mro3] and [Bak] for more details.

12.5 Torsion in Kauffman Bracket Skein Modules In most of the examples mentioned in Sect. 12.4, the Kauffman bracket skein module is torsion-free. Exceptions include the KBSMs of .RP3 # RP3 , .S 2 × S 1 , and more

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generally, .F × S 1 , where F is a closed oriented surface. In fact, the presence of a nonseparating .S 2 in an oriented 3-manifold M always yields torsion in .S2,∞ (M).9 It is enough to use the framing relation to see this. Let M be an oriented 3-manifold with a non-separating .S 2 in it and let L be a framed link intersecting this non-separating 2 . S transversely at exactly one point. The result of ambient isotopy on this link, illustrated in Fig. 12.7, twists its framing twice. We denote this link by . L (2) .Thus, we get that .(A6 − 1)L = 0 in .S2,∞ (M; R, A). This method is often called the “Dirac trick”, “belt trick”, or “light bulb trick”. The light bulb trick and the presence of torsion in the framing skein module will be discussed in more detail in Lecture 15. We note that it is less obvious that the presence of a separating .S 2 in M yields torsion in its Kauffman bracket skein module.

Isotopy

(2)

Isotopy

Isotopy

Isotopy

Isotopy

under

over

Isotopy

Fig. 12.7: The light bulb trick

Conjecture 12.5.1 ([Kir1]) If . M = M1 # M2 , where . Mi is not equal to .S 3 possibly with holes, then .S2,∞ (M) has a torsion element. The conjecture has been proven to hold true partially by the following theorem. Theorem 12.5.2 ([Prz8, Prz10]) 1. If . M1 and . M2 have first homology groups that are not annihilated by 2, that is, .2 · H1 (Mi ; Z) = 0, then the conjecture holds true. 9 Lecture 11 describes more skein modules that exhibit this property.

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2. If there are representations . ρi : π1 (Mi ) −→ SL(2, C) such that the image . ρi (π1 (Mi )) is not in the center of . SL(2, C), then the conjecture holds true. For . M1 = M2 = RP3 , the conjecture was proven in [Mro3]. See Theorem 12.4.18. Theorem 12.5.3 ([Prz15]) Let M be an oriented 3-manifold with an embedded non-separating torus. Then S2,∞ (M) has a torsion element.

.

In [Vev], Michael A. Veve described an example of a 3-manifold with an incompressible torus whose KBSM has torsion, which is detected by the complete hyperbolic structure of the manifold. In particular, he showed that the KBSM of the double of the complement of the figure-eight knot has torsion. In Lecture 15 we will see how torsion arises in the framing skein module. We finish this section with a conjecture by the second author [Bak] about the structure of the KBSM over .Z[A±1 ]. Conjecture 12.5.4 ([Bak]) 1. The KBSM of any closed, prime, oriented 3-manifold can be decomposed into the direct sum of free modules and torsion modules. 2. The KBSM of any closed, irreducible, atoroidal, oriented 3-manifold can be decomposed into the direct sum of free modules and torsion modules. 3. The KBSM of any closed, oriented, irreducible, non-Haken10 3-manifold does not contain torsion and is free. This conjecture follows from a conjecture by the first author in [Kir1] and from Theorem 12.4.15.

12.6 Relative Kauffman Bracket Skein Modules Relative Kauffman bracket skein modules have a well-understood algebraic structure for surface I-bundles and manifolds with marked boundaries, for example, annuli or handlebodies with marked boundaries (see [Prz4, Prz5, Le2]), and such knowledge is especially useful for computing the skein modules of larger 3-manifolds with or without marked boundary. Relative skein modules also allow the theory of Gram determinants (see Lectures 17 and 18) to be developed for 3-manifolds using any skein relation (e.g., the HOMFLYPT skein relation). Definition 12.6.1 Let M be an oriented 3-manifold and .{x1, x2, . . . , x2n } be a set of 2n framed points on .∂ M. Let .L fr (2n) be the set of all relative framed links11 .(M, ∂ M) 10 See Appendix A for definitions of irreducible, atoroidal, and non-Haken 3-manifolds. 11 See Definition 11.8.4 for a description of relative framed links.

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considered up to ambient isotopy keeping .∂ M fixed, such that . L∩∂ M = ∂L = {xi }12n . sub (2n), Let R be a commutative ring with unity, A an invertible element in R, and .S2,∞ the submodule of . RL fr (2n), generated by all the Kauffman bracket skein relations. Then, the relative Kauffman bracket skein module (RKBSM) of M is the quotient: S2,∞ (M, {xi }12n ; R, A) =

.

RL fr (2n) . sub (2n) S2,∞

Remark 12.6.2 In the definition above we allow framed arcs that differ by half twists. In this case, as in the case of the Möbius band framing of closed components, we √ 2 should allow the element . −A (denoted by . A, so . A = −A). Then two links, L and (1/2) , which are identical except that their framings differ by a positive half twist, .L should satisfy the relation . L (1/2) = −(A)3 L in the RKBSM. To exclude the possibility of half-twists on the framing, we can assume that framing is a vector bundle, that is, fibers of annuli and ribbons have orientation. In this case, framed points in the RKBSM should have orientation as well. Compare with Definition 3.4 in [Prz15]. For brevity, we will use the notation .S2,∞ (M, {xi }12n ) when . R = Z[A±1 ]. We now list some useful properties of relative Kauffman bracket skein modules, which are quite similar to those of the usual Kauffman bracket skein modules. Proposition 12.6.3 ([Prz15]) 1. There is a functor from the category of oriented 3-manifolds with 2n framed points on the boundary of the manifolds and orientation-preserving embeddings (up to ambient isotopy fixed on the boundary) to the category of Rmodules with a specified invertible element . A ∈ R. The functor sends an embedding .i : (M, {xi }12n ) c→ (N, {yi }12n ) into the R-module morphism given by 2n 2n .i∗ : S2,∞ (M, {xi } ; R, A) −→ S2,∞ (N, {yi } ; R, A). 1 1 2. Adding a 3-handle to M (outside . xi ) does not change the RKBSM, and adding a 2-handle only adds relations to the RKBSM, since handle slidings yield relations. 3. The RKBSM depends only on the distribution of the framed points . xi among boundary components of M and not on the exact position of these . xi . In particular, if .∂ M is connected, we write .S2,∞ (M, n; R, A) instead of .S2,∞ (M, {xi }12n ; . R, A). 4. The RKBSM satisfies the Universal Coefficient Property. 5. If . M = M1 u M2 is the disjoint union of oriented 3-manifolds . M1 and . M2, then S2,∞ (M, {xi, yi }12n ; R, A) = S2,∞ (M1, {xi }12n ; R, A) ⊗ S2,∞ (M2, {yi }12n ; R, A).

.

12.7 Exercises

245

A particularly useful example of relative Kauffman bracket skein modules arises from considering a thickened disk with 2n framed points on the boundary of the disk and a thickened annulus with 2n framed points on its outer boundary. The theory of RKBSMs of thickenings of surfaces such as the disk and annulus will be discussed in more detail in Lecture 13 and has extensive applications in Lectures 17 and 18.

12.7 Exercises Exercise 12.7.1 Complete the proof of Theorem 12.3.1.3 by showing that if 0 −→ A' −→ A −→ A'' −→ 0 and 0 −→ B ' −→ B −→ B '' −→ 0 are short exact sequences of R-modules, then the following is also a short exact sequence: 0 −→ (A' ⊗ B + A ⊗ B ') −→ A ⊗ B −→ A'' ⊗ B '' −→ 0. Exercise 12.7.2 Express the knot in the annulus, shown in Fig. 12.8, in terms of the basis elements of S2,∞ (Ann × I). Notice that together with the y-axis, this link is the Whitehead link in S 3 (see Fig. 12.8 and Example 12.4.2 ).

(a) Knot in the annulus.

(b) The Whitehead link in

3.

Fig. 12.8: Knot in the annulus, which together with the y-axis is the Whitehead link in S 3 The exercises below support the wrapping conjecture. Exercise 12.7.3 Express the following knot diagrams Dn and Dn' in the standard basis of S2,∞ (F0,2 × I) (see Fig. 12.9). Exercise 12.7.4 Show that w(L) ≥ deg(L) for any link L in Ann × I (see [HP1]). Conjecture 12.7.5 (Wrapping Conjecture) [HP5] Let L be a link in Ann × I and [L] be its class in S2,∞ (Ann × I).12 Then the degree of the polynomial, deg(L), associated to [L] in S2,∞ (Ann × I), is equal to the 12 S2,∞ (Ann × I ) as an algebra is isomorphic to Z[A±1 ][x], where x denotes the homotopically nontrivial curve in the annulus. See Example 13.3.3 in Chapter 13 for details.

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...

...

Fig. 12.9: The twist knot diagrams Dn and Dn' in the annulus wrapping number w(L) of L. Recall that the wrapping number w(L) of a link L is the minimal number of times it intersects with the meridian disk. Definition 12.7.6 Consider a solid torus H1 = F0,2 × I with a fixed meridian disk D2 . We say that a link diagram is annular A-adequate if its A smoothing, Ds A , has no self-touchings (e.g., D is A-adequate) and Ds A intersects the fixed meridian disk the same number of times as the wrapping number of Ds A . We say that a link in a solid torus is annular A-adequate if it has an annulus A-adequate diagram. Similarly, we define an annular B-adequate diagram and link. Exercise 12.7.7 Consider an annular A- or B-adequate diagram of a link L in Ann×I. Let n be the highest degree of L as a polynomial in Z[A±1 ][x].Then show that the wrapping number of L is equal to n. Notice that Dn and Dn' satisfy Conjecture 12.7.5; however, Dn is an alternating and annular A-diagram, while Dn' is neither an A- nor a B-adequate diagram. Exercise 12.7.8 Consider the family of links Ln in H2 = F0,3 ×I illustrated Fig. 12.10. (1) Find Ln in the standard basis of S2,∞ (F0,3 × I). (2) Find the geometric intersection number of Ln with the central compressing disk D2 .

Fig. 12.10: Link diagrams L1 , L2 , and Ln in H2 Exercise 12.7.8 suggests the following.

12.7 Exercises

247

Exercise 12.7.9 Formulate a more general conjecture for thickened surfaces. An example of such a conjecture is given below. Conjecture 12.7.10 Consider the planar surface F0,d , d > 0. Let γ be a properly embedded arc in F0,d not parallel to the boundary and D2 = γ × I be a meridian disk in F0,d × I. Consider any link L in F0,d × I. Then i(L, D2 ) = min j i(b j , D2 ),

.

where i(L, D2 ) is the geometric intersection number of L and D2 , that is, the minimal E 2 crossing number of L and D , and L = j a j b j in the standard basis {b j } of S2,∞ (F0,d × I). fr Exercise 12.7.11 ([Prz8, Prz10]) Consider E the epimorphism, φ : RL → ZH1 (M, comp(L) |Li |, where the sum is taken over all Z), defined by φ(L) = (−1) Li ∈or(L)

possible orientations of L and |Li | is the homology class of an oriented link Li . Furthermore, let φ(Aw) = −φ(w) (i.e., A is sent to −1). Show that φ descends to an homomorphism φˆ : S2,∞ (M) −→ ZH1 (M, Z). Exercise 12.7.12 1. Consider the free involution g : S 1 × S 2 → S 1 × S 2 given by g(z, x) = (z, −x), where z is the complex conjugate of the unit complex number z ∈ S 1 , and −x is the antipodal point of x ∈ S 2 . Show that g is a deck transformation of the covering p : S 1 × S 2 → RP3 # RP3 . 2. Compare S2,∞ (S 1 ×S 2 ) and S2,∞ (RP3 # RP3 ). Do you see any pattern? See [HP5] and [Mro3]. This is an open-ended exercise since there is still no clear picture about the relationship between the KBSMs of covering spaces of 3-manifolds.

Lecture 13 The Kauffman Bracket Skein Module and Algebra of Surface I-Bundles

The Kauffman bracket skein module of a surface times an interval has a natural algebra structure. This algebra has intricate connections with .SL(2, C) character varieties, cluster algebras, and quantum Teichmüller spaces. In this lecture we explore some of these connections and discuss the structure of the Kauffman bracket skein modules and algebras of several thickened surfaces in detail. Relative Kauffman bracket modules, introduced in Lecture 12, of thickened surfaces and their corresponding algebras will also be discussed.

13.1 Introduction Understanding the Kauffman bracket skein module of the product of a surface and an interval is the first step to understanding that of a general 3-manifold. This is due to the fact that any compact oriented 3-manifold can be obtained from a handlebody, which is the product of a planar surface and the unit interval, by adding 2- and 3handles to it (see Appendix A). Moreover, one can project the links in this 3-manifold onto the surface and work with their corresponding link diagrams. This can also be generalized to twisted I-bundles over unoriented surfaces. In Lecture 11 we gave examples of skein modules that have a natural algebra structure. The Kauffman bracket skein module is no different in this aspect. When the 3-manifold is a product of an oriented surface and the interval, the KBSM has a natural algebra structure. However, the Kauffman bracket skein algebra (KBSA) is special in that it forms a bridge between classical low-dimensional topology and quantum topology. In fact, the KBSA is related to the .SL(2, C) character variety, the Witten-Reshetikhin-Turaev topological quantum field theory (Lecture 16 and [BHMV1]), quantum Teichmüller spaces (see [CF, BW, Le4, Kas]), and quantum cluster algebras [Mull]. In this lecture we will discuss how the KBSA is related to the .SL(2, C) character variety and cluster algebras. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_13

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13.2 The Kauffman Bracket Skein Module of Surface I-Bundles We recall and expand the result stated in Example 12.4.2 in Lecture 12 about the Kauffman bracket skein module of I-bundles over surfaces. Theorem 13.2.1 ([Prz4, Prz15, HP4]) × I denote an oriented 3-manifold, which is equal to .F × I, if Let . M = F  F is an oriented surface, and the twisted I-bundle over F, denoted by .F ׈ I, if F is a non-orientable surface. Then .S2,∞ (M; R, A) is a free R-module with basis .B(F) consisting of links (including the empty link .0) in F that have no contractible components. In particular, this result holds for handlebodies since .Hn = F0,n+1 × I, where .Hn is a handlebody of genus n and .Fg,b denotes a surface of genus g with b boundary components. Furthermore, each link in .B(F) is equipped with an arbitrary, but specific, framing. To be concrete, if a link L in F preserves the orientation of F, then we choose as its framing the regular neighborhood of L in F (blackboard framing). If L contains a component K which projects to an orientation reversing curve in F, then its regular neighborhood is a Möbius band and we perform a positive half twist on K to get the framing. It is rather easy to show that .B(F) is a spanning set of .S2,∞ (M; R, A). Every link in M has a regular projection on F and any link can be reduced by the Kauffman bracket skein relations so that the projection has no crossings. The trivial link relation allows us to eliminate trivial components and finally the framing relation allows us to adjust the framing. Thus, .B(F) is a generating set of .S2,∞ (M; R, A). Showing that the elements of .B(F) form a basis for .S2,∞ (M; R, A) is a more laborious task and we refer the reader to [Prz15] for the proof. We note that it is convenient to allow .1/2-framings of links in the proof for twisted I-bundles over a non-orientable surface F. Compare with Remark 12.6.2. Example 13.2.2 .S2,∞ (S 1 × D2 ; R, A) = S2,∞ (F0,2 × I; R, A) and is free and infinitely ∞ , where x denotes the homotopically nontrivial curve generated by the curves .{x i }i=0 0 on the annulus and . x denotes the empty link .0. Furthermore, .S2,∞ (F0,3 × I; R, A) is free and infinitely generated by the monomials .{x i y j z k }i, j,k ≥0 . Here, x, y, and z denote the homotopically nontrivial curves in .F0,3 as illustrated in Fig. 13.2. Note that the empty link is represented by . x 0 y 0 z 0 . Additionally, .S2,∞ (F1,1 × I; R, A) = S2,∞ (F0,3 × I; R, A). See Exercise 13.6.4. Remark 13.2.3 It should be noted that the set .{Si (x)}0∞ also forms a basis for .S2,∞ (F0,2 × I). Here . Si (x) denotes the Chebyshev polynomial of the second kind of degree i (see Lecture 6). Moreover, if 2 is invertible in R, then .{Ti (x)} is also a basis for .S2,∞ (F0,2 × I; R, A), where .Ti (x) denotes the Chebyshev polynomials of the first kind of degree i. For a general R, for .{Ti (x)} to form a basis of .S2,∞ (F0,2 ×I; R, A), the element .2 · 0 must be replaced by .0. This motivated the authors of [Thu] and

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[LTY] to work with a normalized version of Chebyshev polynomials of the first i (x), satisfying .T 0 (x) = 1 and .T i (x) = Ti (x), .i ≥ 1. The result of kind, denoted by .T replacing each component of a link L in .F0,2 by .Ti (x) (or .Si (x)), where x is the core of the annulus, is called threading or decorating L with .Ti (or .Si ). A similar threading procedure for the basis of .S2,∞ (F1,0 × I) is discussed in [FrGe] (see the discussion before Theorem 13.3.6). Example 13.2.4 .S2,∞ (T 2 × I; R, A) is a free R-module generated by the empty link .0, all .(p, q)-curves, and their parallel copies on the torus. These are simple closed curves that wrap along the torus p times in the longitudinal direction and q times in the meridional direction. Here .gcd(p, q) = 1.1 ˆ R, A) = S2,∞ (RP3 ; R, A) = R ⊕ R. The basis of the Example 13.2.5 .S2,∞ (RP2 ×I; KBSM is the empty link .0 and a generator of the fundamental group of .RP3 . Recall ˆ = RP3 \ int(B3 ), and thus, their KBSMs coincide. that .RP2 ×I

13.3 The Kauffman Bracket Skein Algebra of a Surface Times an Interval S2,∞ (F × I; R, A) can be enriched with an algebra structure for which the identity element is the empty link 0. Given two framed links L1 and L2 in F × I, we define their product L1 · L2 by placing L1 over L2 , that is, L1 ⊂ F ×( 12 , 1) and L2 ⊂ F ×(0, 12 ). This multiplication operation is compatible with the module structure, which makes S2,∞ (F × I; R, A) an algebra over R. We denote this algebra by S alg (F; R, A).2 When we switch the order of the factors in the product of two curves in the basis of S alg (F; R, A), the roles of A and A−1 are reversed.3 For brevity, we use the notation S alg (F) when R = Z[A±1 ]. Figure 13.1 illustrates the multiplication of two skeins in S alg (F1,0 ). Remark 13.3.1 1. If A = ±1, then for any 3-manifold M, S2,∞ (M; R, ±1) is an R-algebra. The product L1 · L2 is defined to be their disjoint sum and it does not depend on the relative position of L1 with respect to L2 . This multiplication is commutative and associative and has 0 as the identity. In fact, the algebra only depends on π1 (M). In particular, if h : M −→ N is a homotopy equivalence, then the induced map 1 If .gcd(p, q) p q .( d , d )-curve.

= d = 1, then the .(p, q)-curve on the torus denotes d parallel copies of the

2 The new notation is intended to emphasize the dependence of the product structure on the surface. Indeed, there exist surfaces whose KBSMs are isomorphic but KBSAs are not. See Remark 13.3.1.4. 3 More generally, the roles of A and A−1 are reversed when we take the mirror image L of a link L in F × I .

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13 The Kauffman Bracket Skein Module and Algebra of Surface I-Bundles

=

*

*

=

+

=

=

−1

−1

+

Fig. 13.1: Examples of products of skeins in S alg (F1,0 ) h∗ : S2,∞ (M; R, ±1) −→ S2,∞ (N; R, ±1) is an isomorphism. 2. From Theorem 12.3.5 in Lecture 12 we get that the isomorphism φ of modules is also an isomorphism of algebras φ : S alg (M; R, −1) −→ S alg (M; R, 1). 3. An embedding i : F c→ F ' of oriented surfaces induces a homomorphism i∗ : S alg (F; R, A) −→ S alg (F '; R, A) of their corresponding skein algebras. This leads to a functor from the category of surfaces and embeddings to the category of R-algebras. Compare with Theorem 12.3.1.1 in Lecture 12. 4. Handlebodies do not decompose uniquely as the Cartesian product of surfaces and intervals. It is possible that the products of two non-homeomorphic surfaces F1 and F2 with the unit interval I are homeomorphic, that is, F1 × I = F2 × I, but F1 = F2 (see Exercise 13.6.4). In this case, S2,∞ (F1 ×I; R, A) = S2,∞ (F2 ×I; R, A). However, the algebra structure on the skein module of a thickened surface depends on the product structure. Therefore, S alg (F1 ; R, A) = S alg (F2 ; R, A). The pair of pants and the once punctured torus are two surfaces with this property. Their KBSMs are isomorphic as seen in Example 13.2.2 and Exercise 13.6.4; however, their KBSAs are not. See Examples 13.3.3 and 13.3.4 for a computation of their KBSAs. This fact has interesting ramifications when we categorify the KBSMs of surface I-bundles (see [APS1]). Example 13.3.2 The multiplication operation in S alg (S 2 ) is commutative, since for any two framed links L1 and L2 in S 2 × I, L1 · L2 is ambient isotopic to L2 · L1 . This multiplication corresponds exactly to the usual multiplication of Z[A±1 ]. Thus, S alg (S 2 ) is algebra isomorphic to Z[A±1 ] (compare with Example 12.4.1 in Lecture 12). Example 13.3.3 The basis of S2,∞ (F0,2 × I) is {x i }i ≥0 , where x i represents i parallel copies of the homotopically nontrivial curve x in the annulus (see Fig. 13.2). Thus, S alg (F0,2 ) is the commutative algebra Z[A±1 ][x]. Besides S 2 and the annulus the only

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other surfaces whose KBSAs are commutative are the disk, F0,1 , and the pair of pants, F0,3 . In [BuPr], Doug Bullock and the first author showed that S alg (F0,1 ) = Z[A±1 ], while S alg (F0,3 ) = Z[A±1 ][x, y, z] where x, y, and z represent the three homotopically different boundary parallel curves in F0,3 (see Fig. 13.2). From Remark 13.3.1.1 we see that for any surface F, the KBSA, S alg (F), is always commutative when A = ±1.

(a) Curves in the annulus

(b) Curves in the pair of pants

Fig. 13.2: Basic curves in the KBSAs of the annulus and pair of pants

Example 13.3.4 We now describe the KBSAs of F1,0 and F1,1 . From Fig. 13.1 we see that the multiplication of skeins in S alg (F1,0 ) is not commutative. In [BuPr], Bullock and the first author showed that S alg (F1,0 ) is generated by the curves (1, 0), (0, 1), and (1, 1) on the torus T 2 = F1,0 modulo a set of relations described below. Here (1, 0) represents the longitude of the torus, (0, 1) represents the meridian, and (1, 1) represents the torus curve that travels once each along the longitude and the meridian of the torus. In order to get the set of relations for S alg (F1,0 ), we first compute the relations for S alg (F1,1 ). Consider the following product of curves in S alg (F1,1 ): (1, 0) ∗ (0, 1) = A(1, 1) + A−1 (1, −1) and (0, 1) ∗ (1, 0) = A(1, −1) + A−1 (1, 1).

.

Figure 13.3 illustrates both these products. Therefore, after combining these two equations we get: .

A[(1, 0) ∗ (0, 1)] − A−1 [(0, 1) ∗ (1, 0)] = (A2 − A−2 )(1, 1).

(13.1)

In a similar way one gets: .

A[(0, 1) ∗ (1, 1)] − A−1 [(1, 1) ∗ (0, 1)] = (A2 − A−2 )(1, 0), and .

A[(1, 1) ∗ (1, 0)] − A−1 [(1, 0) ∗ (1, 1)] = (A2 − A−2 )(0, 1).

(13.2) (13.3)

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13 The Kauffman Bracket Skein Module and Algebra of Surface I-Bundles

=

* (1, 0)

=

(0, 1)

(0, 1)

−1

(1, 1)

=

*

+

+

=

(1, 0)

(1, −1)

(1, −1)

−1

(1, 1)

Fig. 13.3: Multiplication of skeins in S alg (F1,1 ) Also, the curve ∂, parallel to the boundary of F1,1 , is generated by (1, 0), (0, 1), and (1, 1), since 2 2 −2 2 .(1, −1) ∗ (1, 1) = A (1, 0) + ∂ + d + A (0, 1) , where d = −A2 − A−2 corresponds to the trivial contractible curve. The lth-power of a curve denotes l parallel copies of it. Equations 13.1, 13.2, and 13.3 constitute the relations for S alg (F1,1 ) and also hold true in S alg (F1,0 ). Moreover, since the boundary curve ∂ becomes contractible in F1,0 , we get the long relation for the KBSA of the torus:4 .

A2 [(1, 1)2 + (1, 0)2 ] + A−2 (0, 1)2 − 2(A2 + A−2 ) − A((1, 0) ∗ (0, 1) ∗ (1, 1)) = 0. (13.4)

Theorem 13.3.5 ([BuPr]) 1. S alg (F1,1 ) =

Z[A±1 ]{(1, 0), (0, 1), (1, 1)} (13.1), (13.2), (13.3)

2. S alg (F1,0 ) =

Z[A±1 ]{(1, 0), (0, 1), (1, 1)} . (13.1), (13.2), (13.3), (13.4)

In [FrGe], Charles Frohman and Răzvan Gelca gave an elegant product-to-sum formula for multiplying curves in S alg (F1,0 ) and used it to show that S alg (F1,0 ) embeds as a subalgebra of the noncommutative torus, which is the q-deformation of the algebra of functions on F1,0 . The elements that enabled them to obtain a neat, compact formula for the multiplication are Chebyshev polynomials of the first kind. For p, q relatively prime denote by (p, q) the (p, q)-curve on the torus. For (p, q) not 4 For A = −1, Eq. 13.4 becomes [(1, 1)2 + (1, 0)2 ] + (0, 1)2 − 4 + ((1, 0) ∗ (0, 1) ∗ (1, 1)) = 0, which is the famous relation discussed by Richard Fricke and Felix Klein in [FrKl].

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255

p q necessarily coprime, define (p, q)T = Tgcd(p,q) ( gcd(p,q) , gcd(p,q) ). This is an element alg of S (F1,0 ) obtained by replacing the variable of the Chebyshev polynomial by the curve on the torus. For example,

(8, 2)T = T2 ((4, 1)) = (4, 1)2 − 2 · 0.

.

Theorem 13.3.6 (The Product-to-Sum Formula [FrGe]) Given two curves (p, q) and (r, s) in T 2 × I, (p, q)T ∗ (r, s)T = t ps−qr (p + q, r + s)T + t −(ps−qr) (p − q, r − s)T .

.

The KBSA of the four-punctured sphere is another example of a noncommutative skein algebra, which was computed in [BuPr] and it is closely related to the KBSA of the torus. In 2018, the authors of [BMPSW] gave a complete description of the multiplicative structure of the KBSA of the thickened four-punctured sphere, essentially finding a product-to-sum formula for infinite families of links in this manifold. Their results will be discussed in more detail in Lecture 14.

2 2 1 1

Fig. 13.4: Curves in F1,2

Example 13.3.7 Consider the surface F1,2 , which is obtained by identifying the edges in Fig. 13.4 top to bottom and left to right, and the curves x1 , x2 , y1 , and y2 in F1,2 as shown. Let a denote the boundary curve in the center of the figure and extend the ground ring to R = Z[A±1 ][a]. Define the curves z1 and z2 by the resolutions x1 y1 = Az1 + A−1 w1 and x2 y2 = Az2 + A−1 w2 in S alg (F1,2 ). Finally, let C(t1, t2, t3 ) denote the set of cyclic commutators {[ti, ti+1 ] A = δti+2 | i = 1, 2, 3}, where the subscripts are taken modulo 3. Theorem 13.3.8 ([BuPr]) S alg (F1,2 ) = R,

.

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where Eq. 13.5 is the following long relation for S alg (F1,2 ): A2 az2 z1 = A2 a2 + A−2 z22 + A6 z12 + (y1 y2 + A4 x1 x2 )a .

− A−1 (x1 y2 + x2 y1 )z2 − (Ax2 y2 + A5 x1 y1 )z1 + x2 y1 x1 y2 +

A6 x12

+

A2 x22

+

A2 y12

+

A−2 y22

(13.5) −2 2

− A (A + A ) . 2

2

A slightly different presentation of this result was given in [Prz10]. Remark 13.3.9 When A = −1, S alg (F0,3 ) = S alg (F1,1 ) and S alg (F1,2 ) = S alg (F0,4 ), since the KBSA depends only on the fundamental group of the surface in this case. Compare with the discussion in Remark 13.3.1.1. Example 13.3.10 S2,∞ (L(2, 1)) has the structure of an algebra. If α denotes an orien\ / A4 − A−4 tation reversing curve in RP2 , then S alg (L(2, 1)) = Z[A±1 ][α]/ α2 − A3 A − A−1 3 (see [Prz10]). Recall that RP is homeomorphic to the lens space L(2, 1) (see Appendix A) and RP3 \ int(B3 ) = RP2 ׈ I . This is one of the very few cases of twisted I-bundles over non-orientable surfaces for which the skein module has an algebra structure.

13.4 Properties of Kauffman Bracket Skein Algebras We now state and discuss some important properties of the Kauffman bracket skein algebra of a surface, including its connection to .SL(2, C) character varieties. From Theorem 13.2.1 we see that if F is not a disk (or .S 2 ), then its KBSM is an infinite dimensional module. Its KBSA, however, is finitely generated. Theorem 13.4.1 ([Bul5, PS3]) If F is a compact oriented surface, then .S alg (F) is finitely generated, and the minimal number of generators is .2rank(H1 (F)) − 1. Adam Sikora and the first author also showed in [PS3] that if . R = Z[A±1, (A2 +A−2 )−1 ] alg and F is ( d )with d holes, then the minimal number of generators of .S (F; R, A) ( d )a disk is .d + 2 + 3 . Theorem 13.4.2 ([PS4]) If R is an integral domain, then .S alg (F; R, A) is Noetherian.

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Theorem 13.4.3 ([PS4]) If . A4n − 1 is not a zero divisor in R for any .n > 0, then the center of .S alg (F; R, A) is a subalgebra generated by the boundary components of F. When A is replaced by a complex root of unity, in addition to the boundary parallel curves, the center of the skein algebra can also contain other skeins, obtained from the threading procedure that uses Chebyshev polynomials of the first kind. Theorem 13.4.4 ([FKL]) Let F be a finite type surface, that is, F is the result of removing finitely many points from a closed, connected, oriented surface, and let A be a root of unity. Then alg (F; C, A) is finitely generated as a module over its center. .S Theorem 13.4.5 ([PS4, Prz16]) 1. .S alg (F; R, A) has no divisors, provided R has no zero divisors. This in turn implies that if R has no nilpotent elements, then neither does .S alg (F; R, A). 2. Let F be an unoriented surface with even negative Euler characteristic and ˆ I. This is the same as saying that M has an even number of projective .M = F × planes as factors. Then .S alg (M; R, ±1) has no zero divisors, provided R has no zero divisors. This result does not hold when F is the Klein bottle. Consider the KBSA of the Klein bottle K b: S alg (K b ׈ I; R, −1) = R[x, y, z]/.

.

If 2 is an invertible element in R, then the right-hand side of the equation above reduces to . R[x, y]/. Thus, it has zero divisors. A sketch of the proof of this theorem was first given in [Prz16]. For a complete proof of part (1) we refer the reader to [PS4].

13.4.1 Connection to the SL(2, C) Character Variety There is a rather surprising and intriguing connection, first noticed by Bullock, between Kauffman bracket skein modules of 3-manifolds and representations of the fundamental groups of the manifolds into the special linear group of .2 × 2 matrices over .C. We give a brief idea of this correspondence here.

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/

\ ab be a matrix in .SL(2, C). Its characteristic polynomial is .λ2 − cd tr(A)λ + 1. From the Cayley-Hamilton theorem we get that . A2 − Atr(A) + Id = 0, or equivalently, . A + A−1 − tr(A) = 0. Then for any . B ∈ SL(2, C), we have −1 B − tr(A)B = 0 and from this we get the fundamental .SL(2, C) trace . AB + A identity −1 .tr(A)tr(B) = tr(AB) + tr(A B) (13.6) Let . A =

with the “initial condition” .tr(Id) = 2. Bullock noticed that this identity is similar to the Kauffman bracket skein relation .[L+ ] = A[L0 ] + A−1 [L∞ ], which for . A = ±1 0 ] = −2. gives .[L+ ] = ±1[L0 ] ± 1[L∞ ] = [L− ] and .[0 This observation led Bullock to consider characters of .SL(2, C) representations of π1 (M) and prove that these characters descend to Kauffman bracket skein modules (e.g., .tr(A) = tr(A−1 ) = a+ d). The next step to formalize this requires Culler-Shalen theory, which shows that sets of .SL(2, C) characters form an algebraic set. We give a precise description of this below following [Bul3] and [Bul2]. In [Bul2], Bullock showed that the character of every representation of .π1 (M) into .SL(2, C) induces a linear functional on a certain specialization of the KBSM of M. In [Bul3], this result was sharpened and it was shown that the KBSM at the specialization, modulo its nilradical, is exactly the ring of characters.

.

Definition 13.4.6 A closed algebraic set X in .Cm is the common zero set of an ideal of polynomials in .C[x1, . . . , xm ]. The elements of .C[x1, . . . , xm ] are polynomial functions on X, and . xi are coordinates on X. The quotient of .C[x1, . . . , xm ] by the ideal of polynomials vanishing on X is called the coordinate ring of X. Definition 13.4.7 An .SL(2, C) representation of the fundamental group of M is a homomorphism . ρ : π1 (M) −→ SL(2, C). The character of a representation is the composition . χρ = tr ◦ ρ (see the following diagram).

Denote the set of all characters by . X(M). For each .γ ∈ π1 (M) consider the function .tγ : X(M) −→ C given by . χρ |−→ χρ (γ). Theorem 13.4.8 ([CS, Vog, FrKl, Hor]) There exists a finite set of elements .{γ1, . . . , γm } in .π1 (M) such that every .tγ is an element of the polynomial ring .C[tγ1 , . . . , tγm ]. For Marc Culler and Peter B. Shalen, this theorem was an initial step toward proving a much deeper result.

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259

Theorem 13.4.9 ([CS]) If every .tγ is an element of .C[tγ1 , . . . , tγm ], then . X(M) is a closed algebraic subset of .C[tγ1 , . . . , tγm ]. Denote the coordinate ring of . X(M) by .R(M). Note that different choices of coordinates lead to different parameterizations of . X(M); however, from [CS] it follows any two parameterizations of . X(M) are equivalent via polynomial maps, and, hence, their coordinate rings are isomorphic. It follows from [CS] that .R(M) lies in .CX(M) , the algebra of functions from . X(M) to .C. → − Each knot K in M determines a unique .tγ as follows. Let . K denote an unspecified orientation on K. Choose any .γ ∈ π1 (M) such that .γ is freely homotopic to an → − embedding of . K . Moreover, the trace is invariant under conjugation and we can → − define . χρ ( K ) = χρ (γ). Since, .tr(A) = tr(A−1 ) in .SL(2, C), we can also define . χρ (K) = χρ (γ). Thus, K determines the map .tγ . Conversely, any .tγ is determined by some (non-unique) K. In [Bul2], Bullock showed the following. Theorem 13.4.10 ([Bul2]) The map .Φ : S alg (M; C, −1) −→ R(M) given by .Φ(K)( χρ ) = − χρ (K) is a well-defined surjective map of algebras. If .S alg (M; C, −1) is generated by the knots .K1, K2, . . . , Km , then .Φ(K1 ), Φ(K2 ), . . ., .Φ(Km ) are coordinates on . X(M). In [Bul3], this result was sharpened and the following theorem was proved. Theorem 13.4.11 ([Bul3])  : CL fr −→ CX(M) be the linear map which sends each knot to the Let .Φ negative of its naturally induced function and each link to the product of the  descends to a well-defined map of algebras images of its components. Then .Φ alg X(M) .Φ : S (M; C, −1) −→ C . The image of this map is the coordinate ring .R(M) of . X(M) and its kernel is the ideal consisting of nilpotent elements of .S alg (M; C, −1).  surjectively descends to .Φ on .S alg (M; C, −1) depends The proof for the fact that .Φ primarily on the observation that the Kauffman bracket skein relation maps to the fundamental .SL(2, C) trace identity described in Eq. 13.6. The characterization of alg (M; C, −1) is more involved and may be found in [Bul3]. .ker φ as the nilradical of .S Furthermore, Theorems 13.4.11 and 13.4.5.1 together imply the following result. Theorem 13.4.12 ([Bul3, PS4]) S alg (F; C, −1) is isomorphic to the coordinate ring of the .SL(2, C) character variety of the fundamental group of the surface. .

One must note that when . A = 1 a similar result holds, but the isomorphism depends on the choice of spin structure on M (see [Bar]). Thus, the KBSM of a

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13 The Kauffman Bracket Skein Module and Algebra of Surface I-Bundles

3-manifold M is a quantization of the coordinate ring of the character variety of the SL(2, C) representations of the fundamental group of M. In [Sik3], Sikora generalized Theorems 13.4.11 and 13.4.12 to .SL(n, C), for any n.

.

13.5 Relative Kauffman Bracket Skein Modules and Algebras of Surface I-Bundles We now discuss the relative Kauffman bracket skein module of surface I-bundles whose definition and basic properties were discussed in Lecture 12. Theorem 13.5.1 ([Prz15]) × I, where .∂F = 0, and let all . xi be marked points that lie on .∂F ×{ 21 }. Let . M = F  Recall that elements of the RKBSM are relative framed links whose endpoints are . xi . Then .S2,∞ (M, {xi }12n ; R, A) is a free R-module whose basis is composed of relative links in F without trivial components. When .n = 0, the empty link is also a generator. Furthermore, each link is equipped with an arbitrary, but specific, framing. We call this basis the standard basis of .S2,∞ (M, {xi }12n ; R, A). The proof of this theorem is the same as that of Theorem 13.2.1 (see [Prz15]); as before relative link diagrams representing the same link are related by Reidemeister × I, {xi }12n ; R, A) moves. When F is a closed surface, the question of whether .S2,∞ (F  2 2 is a free module is still open, except for .F = S and for .F = T and .n = 1, when not all . xi lie on the same boundary component of .F × I (see [Prz15]). Compare with Exercise 13.6.3 and Open Problem 13.5.4. The following corollary to Theorem 13.5.1 uses the language of relative skein modules to formulate well-known combinatorial notions, which have been discussed earlier in Lecture 7. Corollary 13.5.2 ([Prz15]) 1. .S2,∞ (D2 × I, {xi }12n ; R, A) is a free R-module with .Cn = Here .Cn denotes the nth Catalan number.

( ) 1 2n n+1 n

basic elements.

( ) 2. .S2,∞ (Ann×I, {xi }12n ; R, A) is a free . R[x] - module with . Dn = 2n n basic elements, where x denotes the homotopically nontrivial curve in the annulus. The basis is the set of all annular Catalan states which we denote by . Bn . Here, .{xi }12n lie on the outer boundary component of the annulus .Ann. 3. Let x and z denote the core and the boundary parallel curve of the Möbius band, ˆ {xi }12n ; R, A) is a free R-module whose standard basis respectively. .S2,∞ (M b×I,

13.5 Relative Kauffman Bracket Skein Modules and Algebras of Surface I-Bundles

261

is composed of an infinite number of elements of the form .bz i , .bxz i , .i ≥ 0, where . b ∈ Bn , and the finite number of crossingless connections that cut the crosscap ( ) nontrivially. Among them there are . 2n k connections that cut the crosscap .n − k times, for . k < n. See Exercise 7.2.6. ˆ are homeomorphic to the solid torus, and hence, Notice that .Ann × I and . M b×I their RKBSMs are isomorphic as R-modules. However, their standard bases are different since they depend on the I-bundle structure. Corollary 13.5.2 has applications in the study of Gram determinants. See Lectures 17 and 18 for more details.

13.5.1 Connection to Cluster Algebras 2m ) × S 2n There is a natural bilinear map .φ : S2,∞ (F × I; {xi }i=1 2,∞ (F × I; {y j } j=1 ) −→ 2m , {y } 2n ), which is obtained by placing one relative link over S2,∞ (F × I; {xi }i=1 j j=1 another in .F × I. Here . xi = y j for any i and j. This map is also well-defined on relative link diagrams. A small modification of this map leads to an algebra structure on marked diagrams in F, which is defined below.

Let F be an oriented surface with .∂F = 0 and .M = {b1, b2, . . . , bk } be a finite set of points on .∂F, which we call marked points. Consider the RKBSM of .F × I with an arbitrary number of framed points all placed on .{bi } × (0, 1). We denote this version of the RKBSM by .S2,∞ (F × I; M). Notice that .S2,∞ (F × I; M) is the infinite direct sum of the relative Kauffman bracket skein module defined earlier. .S2,∞ (F × I; M) has an algebra structure where the multiplication of two relative links is again defined alg by stacking one relative link over the other. We denote this algebra by .Srel (F; M). To work with projections of links on F, we extend the definition of relative link diagrams to allow simultaneous crossings at the marked points. Additionally, we order the strands in these diagrams. This ordering comes from the height of the framed points in .{bi } × (0, 1) (see Fig. 13.5). When working with such diagrams, we require an additional marked second Reidemeister move (see Fig. 13.6). With this extension of relative link diagrams, we alg can study the relative skein algebra .Srel (F; M) using projections of the relative links on F. Using the Kauffman bracket skein relation and the marked second Reidemeister move, we get the equality illustrated in Fig. 13.7. The relationship between cluster algebras and relative Kauffman bracket skein algebras was first observed by Vladimir Fock and Alexander Goncharov in [FoGo]. In 2012, motivated by examples coming from the theory of cluster algebras and Teichmüller spaces, Greg Muller normalized the definition of relative Kauffman bracket alg skein algebras in [Mull] by taking the quotient of .Srel (F; M) by the submodule generated by the relation illustrated in Fig. 13.8. As a consequence of this, from Fig. 13.7, we get the relation illustrated in Fig. 13.9.

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13 The Kauffman Bracket Skein Module and Algebra of Surface I-Bundles

Fig. 13.5: Fiber with framed points .bink above the marked point .bi ; the region shaded gray represents .F × I while the white region is not a part of the surface

←→

Fig. 13.6: The marked second Reidemeister move

When .F = D2 , .S2,∞ (F × I, {xi }12n ; R, A) can be equipped with another algebra structure which leads to the classical Temperley-Lieb algebra .T Ln (see Lecture 8). Additionally, when F is the annulus, this algebra structure on its RKBSM leads to the annular Temperley-Lieb algebra . AT Ln (see [Jon8, Die]). We end this lecture with a few results about the presence of torsion in relative Kauffman bracket skein modules of thickened surfaces. The result about torsion in the RKBSM of .S 2 × I is left as an exercise to the reader.

13.6 Exercises

263

=

+

−1

Fig. 13.7: Using the second marked Reidemeister moved and Kauffman bracket skein relation.

=

=0

Fig. 13.8: Values of contractible arcs

=

Fig. 13.9: Relation obtained from Figs. 13.7 and 13.8

Theorem 13.5.3 ([Prz15]) S2,∞ (T 2 × I, {xi }12n ) has torsion if not all . xi lie on the same boundary component of .T 2 × I. .

Open Problem 13.5.4 Let F be an oriented closed surface of genus greater than 1 such that not all . xi lie on the same boundary component of .F × I. Does .S2,∞ (F × I, {xi }12n ) have torsion? The answer to this problem can shed some light on the role of incompressible surfaces in the structure of Kauffman bracket skein modules of 3-manifolds.

13.6 Exercises Exercise 13.6.1 Complete the proof of Theorem 13.3.5. Exercise 13.6.2 Show that S alg (F1,0 ; Z, −1) is a unique factorization domain.

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13 The Kauffman Bracket Skein Module and Algebra of Surface I-Bundles

Exercise 13.6.3 Show that if M = S 2 × I and not all xi lie on the same boundary component of M, then S2,∞ (M, {xi }12n ) has torsion. Exercise 13.6.4 Show that Fg,d ×I is homeomorphic to Fg−1,d+2 ×I, for g > 0, d > 0. Hence, show that (1) S2,∞ (Fg,d × I) = S2,∞ (Fg−1,d+2 × I). (2) S alg (Fg,d ) = S alg (Fg−1,d+2 ). Exercise 13.6.5 Verify that the long relation given in Eq. 13.5 holds in S alg (F1,2 ).

−2

0

−1

1

2

Fig. 13.10: Standard basis elements of S2,∞ (F0,2 × I; u, v)

+

=

1

−1

−1

Fig. 13.11: The Kauffman bracket skein relation c0 x = A−1 c1 + A−1 c−1

Exercise 13.6.6 Consider two marked points u and v on the boundary of the annulus as illustrated in Fig. 13.10. Show that (1) The arcs {ci }i ∈Z illustrated in Fig. 13.10 form the standard basis of S2,∞ (F0,2 × I; u, v). (2) The elements {c0 x i }, {c1 x i }, i ≥ 0 form another basis of S2,∞ (F0,2 × I; u, v), where x is the homotopically nontrivial curve on the annulus. See Fig. 13.11.

Lecture 14 Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened T-Shirt

In 2018, along with Sujoy Mukherjee, Marithania Silvero, and Xiao Wang, the first two authors studied the multiplicative structure of the Kauffman bracket skein algebra (KBSA) of the four-holed sphere also known as the thickened T-shirt. Generalizing the product-to-sum formula given by Charles Frohman and Răzvan Gelca for the Kauffman bracket skein algebra of the torus, they presented an algorithm to compute the product of any two elements of the algebra and gave an explicit formula for some infinite families of curves. Further, they conjectured the existence of a positive basis for the algebra. In this lecture we discuss their work in detail and briefly touch upon Pierrick Bousseau’s work in resolving this conjecture.

14.1 Introduction In this lecture we discuss the Kauffman bracket skein algebra of the thickened T-shirt. Skein modules were introduced in 1987 by the first author (see [Prz4, Prz15] and Lecture 12) as invariants of codimension two embeddings of manifolds, in particular of 1-manifolds embedded in 3-manifolds. They were independently introduced by Vladimir Turaev in 1988 [Tur3]. The Kauffman bracket skein module and algebra are generalizations of the Kauffman bracket polynomial of links in the 3-sphere. Certain skein modules, like the KBSM of the trivial bundle over an oriented surface, can be endowed with an algebra structure by defining a multiplication operation between the elements of the module. See Lecture 13 for a description of the Kauffman bracket skein algebra. In 1997, Charles Frohman and Răzvan Gelca discovered a famous product-to-sum formula for the multiplication of elements in the KBSA of the thickened torus and showed that this algebra is isomorphic to a subalgebra of the noncommutative torus. They achieved this by decorating the basis of the KBSA with Chebyshev polynomials. We refer the reader to Lecture 13 and [FrGe] for these results. Chebyshev polynomials have been discussed in Lecture 6. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_14

265

266

14 Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened

One of the features of the Chebyshev basis for .S alg (T 2 ) is that it is positive. A basis .B for .S alg (F) over .Z[A±1 ] is said to be positive if the structure constants for multiplication lie in .Z ≥0 [A±1 ], that is, for any . xi and . x j in .B their product E . xi · x j = ak xk , where the structure constants .ak ∈ Z ≥0 [A±1 ] and . xk ∈ B. The following question arises for Kauffman bracket skein algebras. Open Question Which bases of the KBSA of trivial bundles over oriented surfaces are positive? This question has intrigued skein theorists for more than a decade and was first posed by Vladimir Fock and Alexander Goncharov [FoGo] in 2006 while working with cluster algebras. In 2013, Dylan Thurston [Thu] conjectured the following.1 Conjecture 14.1.1 ([Thu]) The Chebyshev basis for .S alg (F; R, A) has positive structure constants.2 He proved this conjecture for the special case when . R = Z and . A = 1. Motivated by Frohman and Gelca’s work and the positivity conjecture, the authors of the paper [BMPSW] provided an algorithm to compute the product of any two elements in the skein algebra of the four-holed sphere3 and give explicit formulas for several infinite families of curves. The algorithm is based on the action of the generators of the mapping class group of .F0,4 on the basis elements of .S alg (F0,4 ) and uses properties of the Farey diagram along with an adapted version of the Euclidean algorithm. In this lecture we discuss this algorithm in detail. The KBSA of the four-holed sphere .F0,4 is closely related to that of the torus since .T 2 is the double cover of .S 2 branched along four ramification points. The deck transformation of this covering is called a hyperelliptic involution of the torus. See Fig. 14.1 for an illustration.

14.1.1 Chebyshev Decoration Recall from Lecture 13 that multicurves on the torus are parameterized by the pairs (d, n) ∈ Z+ × Z, often denoted by unreduced fractions . dn , where .Z+ denotes the set of all nonnegative integers. The number of components of this multicurve is equal n/r is called the slope of the multicurve. to .r = gcd(d, n) and the reduced fraction . d/r

.

1 A proof of this conjecture was announced by Hoel Queffelec in [Que] after the final draft of this book had been submitted. 2 To get a basis for . S alg (F; R, A), the version of Chebyshev polynomials of the first kind used -n (x), satisfy .T -0 (x) = 1 and by Thurston is slightly adjusted. These polynomials, denoted by .T -n (x) = Tn (x), .n ≥ 1. .T 3 The four-holed sphere is often called a T-shirt. We also note that we can work with the fourpunctured sphere since the KBSM does not distinguish punctures from holes.

14.1 Introduction

267

•(

)

(−

− )•

Double Branched Covering

Double Branched Covering

•(

)

Fig. 14.1: The double branched cover .F1,0 −→ S 2 branched along four points. The deck transformation is hyperelliptic involution

By convention, we denote the curve .(−d, n) by .(d, −n), .d > 0, and the empty link by (0, 0).

.

We now describe a method of drawing curves on .F0,4 . Define the pair .(d, n) ∈ Z+ × Z (also denoted by . dn ) to be the multicurve whose (minimal) intersection numbers with the x-axis and the y-axis are .2|n| and 2d, respectively. The sign of n is given by the direction of the curve. There are many ways of drawing curves of given slopes in .F0,4 . Figure 14.2 illustrates one such way of drawing the curve .(d, n), n > 0, in . F0,4 × I. If .n < 0, one should take the reflection of the curve .(d, |n|) about the vertical axis (see [BMPSW] for more details). This is a special case of Dehn coordinates of multicurves in any oriented surface with negative Euler characteristic (see [Deh3, PH] for a detailed description). As in the case of the torus .(d, n) on . F0,4 decorated by the (see Lecture 13), we denote by .(d, n)T the multicurve ) ( d n , gcd(d,n) . Similarly, .(d, n)S Chebyshev polynomial of the first kind, .Tgcd(d,n) gcd(d,n) denotes the multicurve( .(d, n) on .F0,4 decorated by the Chebyshev polynomial of the )

second kind, .Sgcd(d,n)

n d gcd(d,n) , gcd(d,n)

.

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14 Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened

Fig. 14.2: The curve .(3, 7) = 37 is drawn as follows. Start with a .3 × 7 grid as shown in (a). The vertices of the rectangle represent the 4 punctures in .S 2 . We draw an arc starting at the lower left corner of the “pillowcase” in (b). The arc in (c) is the core of the curve .(3, 7) shown in (d)

14.2 Structure of the KBSA of the Four-Punctured Sphere In this section we give the exact structure of the KBSA of the thickened fourpunctured sphere using generators and relations. The KBSA .S alg (F0,4 ) is generated by the curves .(1, 0), (0, 1), and .(1, 1) together with the curves .ai , .i = 1, 2, 3, 4, where the curves .ai are parallel to the punctures . pi (see Fig. 14.2a). Recall from Lecture 13 that the boundary curves .ai generate the center of .S alg (F0,4 ). Consider the following product in .S alg (F0,4 ) of curves .(1, 0) and .(0, 1), illustrated in Fig. 14.3: (1, 0) ∗ (0, 1) = A2 (1, 1) + a1 a3 + a2 a4 + A−2 (1, −1).

.

=

(1, 0) * (0, 1)

2

+

(1, 1)

+

1 3

+

2 4

Fig. 14.3: The product .(1, 0) ∗ (0, 1) in .S alg (F0,4 ) Therefore,

−2

(1, −1)

14.3 The Action of PSL(2, Z) on Multicurves in the Thickened T-Shirt

269

(0, 1) ∗ (1, 0) = A2 (1, −1) + a1 a3 + a2 a4 + A−2 (1, 1).

.

From these equations we obtain the following: .

A2 (1, 0)∗(0, 1)− A−2 (0, 1)∗(1, 0) = (A4 − A−4 )(1, 1)+(A2 − A−2 )(a1 a3 +a2 a4 ). (14.1)

Similarly, by considering the products .(0, 1) ∗ (1, 1) and .(1, 0) ∗ (1, 1), we get the following two equations: .

A2 (0, 1)∗(1, 1)− A−2 (1, 1)∗(0, 1) = (A4 − A−4 )(1, 0)+(A2 − A−2 )(a1 a4 +a2 a3 ). (14.2)

.

A2 (1, 1)∗(1, 0)− A−2 (1, 0)∗(1, 1) = (A4 − A−4 )(0, 1)+(A2 − A−2 )(a1 a2 +a3 a4 ). (14.3)

By considering the product of the curves .(1, 0), .(0, 1), and .(1, 1), we obtain the long relation for the KBSA of .F0,4 × I: A4 [(1, 1)2 + A2 (1, 0)2 ] + A−4 (0, 1)2 + A2 [(a1 a4 + a2 a3 )(1, 0) + (a1 a3 + a2 a4 )(1, 1)] .

+ A−2 (a1 a2 + a3 a4 )(0, 1) + [a12 + a22 + a32 + a42 + a1 a2 a3 a4 + (A2 − A−2 )2 ] − 2(A4 + A−4 ) − A2 (1, 0) ∗ (0, 1) ∗ (1, 1) = 0 (14.4) The four equations listed above are sufficient to describe the KBSA of .F0,4 × I.

Theorem 14.2.1 ([BuPr]) S alg (F0,4 ) =

.

Z[A±1 ][a1, a2, a3, a4 ] {(1, 0), (0, 1), (1, 1)} . (14.1), (14.2), (14.3), (14.4)

Corollary 14.2.2 Let . R0,1 = a1 a2 + a3 a4 , . R1,0 = a1 a4 + a2 a3 , . R1,1 = a1 a3 + a2 a4 and . y = a12 + a22 + a32 + a42 + a1 a2 a3 a4 + (A2 − A−2 )2 . Then, modulo . R0,1, R1,0, R1,1 and y, the relations associated to the KBSA of .F1,0 and .F0,4 coincide up to switching A and . A2 and keeping the slopes of the curves unchanged. See Lecture 13 for the structure of .S alg (F1,0 ). Corollary 14.2.2 suggests that when looking for product-to-sum formulas for the multiplication of two curves in .F0,4 , one should work with a subring of the center of .S2,∞ (F0,4 × I) generated by the variables . R0,1, R1,0, R1,1 , and y. We denote this subring by .K0 .

270

14 Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened

14.3 The Action of PSL(2, Z) on Multicurves in the Thickened T-Shirt The algorithm to algebraically compute the product of closed multicurves in .F0,4 × I depends on the action of .PSL(2, Z) on multicurves in .F0,4 . We describe this action in this section. The mapping class group of the torus is .Mod+ (T 2 ) = SL(2, Z) (see [Deh3]), and that of the four-punctured sphere is the semidirect product .PSL(2, Z) x (Z2 × Z2 ) (we allow permutations of the punctures, compare with [FaMa] and see Appendix B). The action of .PSL(2, Z) on .Q ∪ 10 is encoded by the Farey diagram, which is illustrated in Fig. 14.4. The following is a short description of this action (see [Hat3] for more detail).

Fig. 14.4: The Farey diagram The Farey diagram consists of vertices indexed by fractions .Q ∪ 10 , where vertices and . dc are connected by an edge if and only if the determinant .(ad − bc) of the [ ] ac matrix . is .±1. The vertices of the Farey diagram are labeled according to the bd following scheme: if the labels at the ends of the long edge of a triangle are . ba and

a . b

14.4 Product-to-Sum Formula for Some Families of Curves in the Thickened T-Shirt

271

then the label on the third vertex of the triangle is the fraction . a+b c+d . This fraction is known as the mediant of . ba and . dc . Farey diagrams have been used extensively in the study of low-dimensional topology. See for example, [HT, FH], and [Prz30]. [ ] 11 .PSL(2, Z) acts on the curves on . F0,4 as follows: the generators . s1 = and 01 [ ] 10 . s2 = act on the fractions . dn by left matrix multiplication. The generators act 11 on the center by switching .a3 with .a4 and .a1 with .a4 , respectively. Therefore, under the action of .s1 , . R1,1 goes to . R1,0 , . R0,1 goes to . R1,1 , and . R0,1 remains unchanged. The second generator, .s2 , sends . R1,1 to . R0,1 and . R0,1 to . R1,1 and keeps . R1,0 fixed. The variable y remains unchanged by the action. .

c d,

To summarize, in preparation for discussing the algorithm, we have described a way to draw curves in .F0,4 using Dehn coordinates, defined the Chebyshev decoration for multicurves in .F0,4 , and described the structure of the KBSA of .F0,4 × I, and we have elucidated the action of .PSL(2, Z) on curves in .F0,4 which is given by the Farey diagram. We now turn our attention to a few key results, which constitute the initial data of the algorithm given in [BMPSW] to multiply curves in .F0,4 × I.

14.4 Product-to-Sum Formula for Some Families of Curves in the Thickened T-Shirt In this section we give closed formulas for the product of some families of curves in .F0,4 yielding relatively simple formulas. In the following subsection we provide some basic computational results for the multiplication of two multicurves when the absolute value of their associated determinant is either 0, 1, or 2. These will serve as the initial data for the algorithm presented in Sect. 14.5.

14.4.1 Basic Formulas and Initial Data Given the product of two multicurves .(d1, n1 ) ∗ (d2, n2 ), we associate to it the determinant .d1 n2 − d2 n1 . This determinant is related to the minimal intersection number of the curves .(d1, n1 ) and .(d2, n2 ) (see Lemma 14.5.1). Definition 14.4.1 The geometric intersection number between free homotopy classes of multicurves in a surface F is defined to be the minimal number of intersection points between representative multicurves in the two homotopy classes.

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14 Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened

Consider the product .(d1, n1 ) ∗ (d2, n2 ) with determinant .d1 n2 − d2 n1 = 0, then we have .(d1, n1 ) ∗ (d2, n2 ) = (d1 + d2, n1 + n2 ). From the product-to-sum formula for Chebyshev polynomials described in Lecture 13, we have .(d1, n1 )T ∗ (d2, n2 )T = (d1 + d2, n1 + n2 )T + (d1 − d2, n1 − n2 )T . Lemma 14.4.2 ([BMPSW]) Consider two curves in .F0,4, .(d1, n1 ) = n2 d2 , satisfying . d1 n2 − d2 n1 = 1. Then

n1 d1

and .(d2, n2 ) =

(d1, n1 ) ∗ (d2, n2 ) = A2 (d1 + d2, n1 + n2 ) + A−2 (d1 − d2, n1 − n2 ) + Rd1 +d2,n1 +n2 ,

.

where the indices of R are taken modulo 2. In particular we have: (i) .(1, 0) ∗ (1, 1) = A2 (2, 1) + A−2 (0, 1) + a1 a2 + a3 a4 = A2 (2, 1) + A−2 (0, 1) + R0,1, (ii) .(1, 0) ∗ (0, 1) = A2 (1, 1) + A−2 (−1, 1) + a1 a3 + a2 a4 = A2 (1, 1) + A−2 (−1, 1) + R1,1, and (iii) .(1, 1) ∗ (0, 1) = A2 (1, 2) + A−2 (1, 0) + a1 a4 + a2 a3 = A2 (1, 2) + A−2 (1, 0) + R1,0 . Proof It suffices to prove one of cases (i), (ii), or (iii). The lemma follows by applying the action of .PSL(2, Z) on the basic elements in the formulas. The calculation for case (ii) is presented in Fig. 14.3. u n Lemma 14.4.3 ([BMPSW]) Let .(d1, n1 ), (d2, n2 ) be two curves in .F0,4 so that gcd(d1, n1 ) = gcd(d2, n2 ) = 1 and .d1 n2 − d2 n1 = 2. Then,

.

(d1, n1 ) ∗ (d2, n2 ) = A4 (d1 + d2, n1 + n2 )T + A−4 (d1 − d2, n1 − n2 )T + y + A2 R d1 +d2 , n1 +n2 ((d1 + d2 )/2, (n1 + n2 )/2)

.

2

2

−2

+ A R d1 −d2 , n1 −n2 ((d1 − d2 )/2, (n1 − n2 )/2). 2

2

In particular, we have .(1, −1) ∗ (1, 1) = A4 (2, 0)T + A−4 (0, 2)T + y + A2 R1,0 (1, 0) + A−2 R0,1 (0, 1). Proof As in the previous lemma, it suffices to find the formula for .(1, −1) ∗ (1, 1). u n Calculations for this case are shown in Fig. 14.5. Lemma 14.4.4 ([BMPSW]) Let .(d1, n1 ), (d2, n2 ) be two multicurves in .F0,4 so that d1 n2 − d2 n1 = 2.

.

1. If .gcd(d1, n1 ) = 2, then (d1, n1 )T ∗ (d2, n2 ) = A4 (d1 + d2, n1 + n2 ) + A−4 (d1 − d2, n1 − n2 ) .

+ (d1 /2, n1 /2)R d1 +d , n1 +n + (A2 + A−2 )Rd2,n2 . 2

2

2

2

14.4 Product-to-Sum Formula for Some Families of Curves in the Thickened T-Shirt

= (1, −1

4

+

+

(1, 1) 2

* 1, 1) + 2 1

−2

+

+

2 3 (1, 0)

+

+

2

+

1 4 (1, 0)

+

+

+

2 2

2 3

2 4

2

+

+

+

+

1 2 (0, 1)

273

3 4 (0, 1)

1 2 3 4

−4

(0, 1) 2

Fig. 14.5: The product .(1, −1) ∗ (1, 1) in .S alg (F0,4 ) 2. If .gcd(d2, n2 ) = 2, then (d1, n1 ) ∗ (d2, n2 )T = A4 (d1 + d2, n1 + n2 ) + A−4 (d1 − d2, n1 − n2 ) .

+ R d1 +d2 , n1 +n2 (d2 /2, n2 /2) + (A2 + A−2 )Rd1,n1 . 2

2

Proof We prove case .(1) by applying Lemma 14.4.2 twice: ( ( ) ) ( ) n1 /2 n2 n1 n2 n2 n1 /2 ∗ ∗ (d1, n1 )T ∗ (d2, n2 ) = ∗ = −2 d1 T d2 d1 /2 d1 /2 d2 d2 ( ) ( ) n2 Lemma 14.4.2 n1 /2 2 n1 /2 + n2 −2 n2 − n1 /2 ∗ A + R d1 +d , n1 +n − 2 +A ========= 2 2 d1 /2 d1 /2 + d2 d2 − d1 /2 d2 2 2 ( ) n1 + n2 n2 Lemma 14.4.2 ========= A2 A2 + A−2 + Rd2,n2 d1 + d2 d2 . ( ) −2 n2 − n1 −2 2 n2 +A +A + Rd2,n2 A d2 d2 − d1 ( ) n2 n1 /2 R d1 +d , n1 +n − 2 + 2 2 d1 /2 2 d2 2 n − n n + n n1 /2 1 2 2 1 Rd + (A2 + A−2 )Rd2,n2 . + A−4 + = A4 n d1 + d2 d2 − d1 d1 /2 21 +d2, 21 +n2 The proof of case .(2) is analogous.

u n

274

14 Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened

Remark 14.4.5 After interchanging the roles of A and . A−1 , Lemma 14.4.2 holds when the determinant of the curves is equal to .−1 and Lemmas 14.4.3 and 14.4.4 hold when the determinant of the curves is equal to .−2.

14.4.2 Two Closed Formulas In this subsection we present formulas for the product of two infinite families of curves with the curve .(0, 1). Theorem 14.4.6 ([BMPSW]) Let .m > 0. Then, (m, 0)T ∗ (0, 1) = A2m (m, 1) + A−2m (m, −1) + (m − 1, 0)S R1,1 .

+

m−1 E

(A2i + A−2i )(m − 1 − i, 0)S Ri+1,1 .

i=1

Proof For .m = 1 and 2 the formula follows from Lemmas 14.4.2 and 14.4.4. We now assume that the formula holds up to m (.m > 2) and proceed by induction on m by applying the recurrence relation of Chebyshev polynomials:

Chebyshev

(m + 1, 0)T ∗ (0, 1) ======= (1, 0) ∗ (m, 0)T ∗ (0, 1) − (m − 1, 0)T ∗ (0, 1) ======= (1, 0) ∗ [A2m (m, 1) + A−2m (m, −1)] − [A2m−2 (m − 1, 1) Induction

+ A−2m+2 (m − 1, −1)] + (1, 0) ∗ (m − 1, 0)S R1,1 − (m − 2, 0)S R1,1 [ m−1 ] E 2i −2i + (1, 0) ∗ (A + A )(m − 1 − i, 0)S Ri+1,1 i=1

.

−

m−2 E

(A2i + A−2i )(m − 2 − i, 0)S Ri+1,1

i=1 Det=±1,Chebyshev

========

A2m+2 (m + 1, 1) + A2m−2 (m − 1, 1) + A2m Rm+1,1 + A−2m Rm+1,1

+ A2−2m (m − 1, −1) + A−2m−2 (m + 1, −1) − A2m−2 (m − 1, 1) − A2−2m (m − 1, −1) + (m, 0)S R1,1 +

m−1 E

(A2i + A−2i )(m − i, 0)S Ri+1,1

i=1

= A2m+2 (m + 1, 1) + A−2m−2 (m + 1, −1) + (m, 0)S R1,1 m E + (A2i + A−2i )(m − i, 0)S Ri+1,1, i=1

14.5 The Algorithm

275

u n

as needed.4

The following theorem gives the formula for the product of curves in the family (n, 1) ∗ (0, 1). A proof may be found in [BMPSW].

.

Theorem 14.4.7 ([BMPSW]) Let n be a nonnegative integer. Then, (n, 1) ∗ (0, 1) = A2n (n, 2)T + A−2n (n, 0)T +

n−1 E

.

αn−i βi + A2

i=0

{

where .

.

αi =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

R1,0 y R0,1 R1,1 + R1,0 (i−2) 2 2 2 (R0,1 + R1,1 ) (i − 2)R0,1 R1,1

i i i i i

ni =

n−1 E

[ni ] A4 Ri−n,1 (i, 1),

i=1

i for i ≤ n2 , n − i for i ≥ n2 ,

⎧ (i−1)/2 = 1, ⎪ E ⎪ −2i+8j (i − 2 j, 0) i odd, ⎪ S ⎪ = 2, ⎨ j=0 A ⎪ . and β = i = 3, i/2 E −2i+8j ⎪ ⎪ ⎪ A (i − 2 j, 0)S i even. ⎪ > 3 even, ⎪ j=0 ⎩ > 3 odd,

Notice that all the coefficients are positive and unimodal. Here .[n]q = 1 + q + q2 + ... + q n−1 denotes the nth quantum integer (see Lecture 9).

14.5 The Algorithm We now analyze the algorithm which, given an ordered pair of curves in .F0,4 × I, returns their product as an element of .S alg (F0,4 ) in terms of the generators in the basis. The first step in the algorithm consists of reducing the product .(d1, n1 )∗(d2, n2 ) to the form .(d, n) ∗ (0, k), with .0 ≤ n < d and . k > 0. This is achieved by using the action of .PSL(2, Z) on .F0,4 together with an adapted version of the Euclidean algorithm. The following lemmas serve as a preparation for the second step [ ] of the algorithm. n2 n1 with the geometric Lemma 14.5.1 relates the determinant (.d1 n2 − d2 n1 ) of . d2 d1 intersection number of a pair of multicurves .(d1, n1 ) and .(d2, n2 ). Lemma 14.5.2 is used to show the correctness of the algorithm when . k > 1 or .gcd(d, n) > 1. Lemmas 14.5.3 and 14.5.4 are used in Theorem 14.5.5 to prove the correctness of the algorithm when . k = 1 and .gcd(d, n) = 1.

4 Notice that we use the fact that the Chebyshev polynomials .Tn and .Sn have the same recursive relation.

276

14 Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened

Lemma 14.5.1 ([BMPSW]) Consider any two simple closed multicurves .(c1 d1, c1 n1 ) and .(c2 d2, c2 n2 ) with gcd(d1, n1 ) = 1 = gcd(d2, n2 ) in .F0,4 × I, where .ci, di, ni ∈ Z and .i = 1, 2. Then their geometric intersection number is equal to .2|c1 d1 c2 n2 − c1 n1 c2 d2 | = 2|c1 c2 ||d1 n2 − n1 d2 |. See Sections 1.2.3 and 2.2.5 in [FaMa].

.

Lemma 14.5.2 ([BMPSW]) Consider .(c1 d1, c1 n1 ) ∗ (c2 d2, c2 n2 ) with .gcd(d1, n1 ) = 1 = gcd(d2, n2 ). Without loss of generality, assume .c1 > c2 ≥ 1. Suppose that .((c1 − 1)d1, (c1 − 1)n1 ) ∗ E (c2 d2, c2 n2 ) = i ui · (qi, pi ), .ui ∈ K0 (see the discussion after Corollary 14.2.2). Then we have . |d1 pi − n1 qi | < |c1 d1 c2 n2 − c1 n1 c2 d2 |. Proof The curves .(qi, pi ) are obtained after smoothing of all the crossings of .(c1 − 1) parallel copies of the curves .(d1, n1 ) and .c2 parallel copies of the curves .(d2, n2 ). During this process, we observe that the number of crossings, say n, of the remaining copy of the curve .(d1, n1 ) with .(qi, pi ) does not increase. Thus, .n = 2|d1 pi − n1 qi | ≤ u n 2|d1 c2 n2 − n1 c2 d2 | ≤ |c1 d1 c2 n2 − c1 n1 c2 d2 | < 2|c1 d1 c2 n2 − c1 n1 c2 d2 |. Lemma 14.5.3 ([BMPSW]) Consider a curve .(d, n) (.d ≥ 0) and let .(d, n) ∗ (0, 1) = Then,

E i

ui · (qi, pi ), with .qi ≥ 0.

(1) . |pi | ≤ n + 1 and .qi ≤ d. (2) . |pi d − qi n| ≤ d. (3) If .n > 0, then . pi ≥ 0. Proof 1. Observe that the coordinate lines cannot intersect a .(qi, pi ) curve more than the number of times the curves .(d, n) ∪ (0, 1) intersect the .(qi, pi ) curve. 2. The curve .(d, n) can intersect any curve resulting from the product .(d, n) ∗ (0, 1) (including .(qi, pi )) in no more than 2d points. Thus, by Lemma 14.5.1, the statement follows. 3. This case follows from (2) since we have .−d ≤ pi d − qi n. So . dn qi − 1 ≤ pi and, therefore, for .n > 0 and .qi > 0 we have . pi > −1. If .qi = 0, then we can assume . pi ≥ 0 by convention. u n Lemma 14.5.3 is generalized in Theorem 14.5.7. Lemma 14.5.4 ([BMPSW]) (1) Consider the fractions . nd11 ,. dn22 , and . qp with positive denominators. Then

14.5 The Algorithm

277 .

|n2 q − d2 p| ≤

d2 q |n2 d1 − n1 d2 | + |n1 q − d1 p|. d1 d1

(2) In particular, if . |n2 d1 − n1 d2 | = 1, . |n1 q − d1 p| ≤ d1 , and .q ≤ d1, then .

|n2 q − d2 p| ≤ 1 + d2 .

Proof The inequality in (1) is essentially the triangle inequality for fractions (rational numbers). We see this immediately when we divide the two sides of the inequality in (1) by a positive number .qd2 . We obtain the inequality: | | | | | | | n2 p | | n2 n1 | | n1 p | | | |. | | | | d2 − q | ≤ | d2 − d1 | + | d1 − q | u n

which is just a triangle inequality of rational numbers. Theorem 14.5.5 ([BMPSW])

Consider the product .(d, n) ∗ (0, 1) with .1 < n < d ≥ 3 and .gcd(d, n) = 1. Given products of pairs of curves with determinant less than d, the product .(d, n) ∗ (0, 1) can be computed. The procedure for the computation is elucidated in the proof. Proof Let .n, d > 1, .gcd(n, d) = 1, and assume that the reduced fractions . nd11 and n2 n1 n2 n n . d2 “support" . d in the Farey diagram, that is, . d is the mediant of . d1 and . d2 . Thus, .n = n1 + n2 , . d = d1 + d2 , and . |n2 d1 − n1 d2 | = 1. Then . d1, d2 and .n1, n2 satisfy −2(d1 n2 −d2 n1 ) (d , n ) ∗ .0 < d2 < d, .0 < d1 < d − 1, .0 < n1, n2 . Thus, .(d, n) ∗ (0, 1) = A 1 1 −2(d n −d n ) −4(d z−d n ) 1 2 2 1 1 2 1 Rd,n (0, 1) − A (d1 − d2, n1 − n2 ) ∗ (0, 1). (d2, n2 ) ∗ (0, 1) − A For .(d1 − d2, n1 − n2 ) ∗ (0, 1), we know the product since the absolute value of the determinant is . |d1 − d2 | < d. So we E focus on .(d1, n1 ) ∗ (d2, n2 ) ∗ (0, 1). Assume .(d1, n1 ) ∗ (d2, n2 ) ∗ (0, 1) = (d1, n1 ) ∗ i ui · (qi, pi ) as in Lemma 14.5.3 (1). By parts (1) and (2) of Lemma 14.5.3 and part (2) of Lemma 14.5.4, we have . |n1 pi − d1 pi | ≤ u n 1 + d1 < d. We summarize the above results in the following theorem, which proves the correctness of the algorithm. Theorem 14.5.6 ([BMPSW]) Consider the product .(d, n) ∗ (0, k) with .0 ≤ n ≤ d > 0, . k ≥ 0. One can compute the product algebraically with knowledge of the products of .(d1, n1 ) ∗ (d2, n2 ) with . |d1 n2 − d2 n1 | < dk. Proof If . k = 0, .(d, n) ∗ (0, 0) = (d, n). If . k > 1, by Lemma 14.5.2, the statement is true. Now, consider the case . k = 1. We can assume that .d = ms + r, .n = s, and we multiply .(ms + r, s) ∗ (0, 1) as before to get E 2(ms+r) .(ms +r, s) ∗ (0, 1) = A (ms +r, s + 1) + A−2(ms+r) (ms +r, s − 1) + ui · (qi, pi ). i

278

14 Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened

If .n = 1, we get the closed formula in Theorem 14.4.7. If .n > 1, then we have the following cases: (1) When .gcd(d, n) > 1, by Lemma 14.5.2, the statement is true. (2) When .gcd(d, n) = 1 and .d ≥ 3, by Theorem 14.5.5, the statement is true. The remaining finite cases for smaller determinants have been checked in Sect. 14.4 and in Lemmas 14.4.2, 14.4.3, and 14.4.4. u n This completes the algorithm. The next theorem generalizes Lemma 14.5.3. Theorem 14.5.7 ([BMPSW]) Consider the product .(d, n) ∗ (0, 1) with .0 ≤ n ≤ d > 0. Then, E 2d −2d .(d, n) ∗ (0, 1) = A (d, n + 1) + A (d, n − 1) + ui · (qi, pi ), i

with .0 ≤ qi ≤ d − 1 and .0 ≤ pi ≤ n. Proof Consider a curve (or multicurve) of slope type . dn with .n, d > 0. In Dehn coordinates5 . dn has coordinates .(2d, n), that is, there are 4d points on a Dehn annulus (i.e., the annulus along a pants curve, which is the chosen curve that decomposes . F0,4 into two pairs of pants), say . x1, x2, . . . , x2d on one boundary component and n 1 . y1, y2, . . . , y2d on the other. Now consider the diagram . d ∪ 0 (which is the product before applying skein relations). We can take as the . 10 curve the boundary of the pants annulus that contains the points . xi . Every generic horizontal (slope . 01 ) curve, say h, intersects . dn ∪ 10 in .2n + 2 points. We can always deform h to a curve, say ' . h , which intersects the crossing, say at the point . xi , and is otherwise generic and intersects . dn ∪ 10 in 2n additional points (there are always two choices: if . xi is one point, then . x2d+1−i is the other). If we smooth the crossing . xi in B type (along a B marker), then the diagram obtained by this smoothing is intersected by . h ' in 2n points. Furthermore, for any crossing . xi we can find its h type curve. Therefore, the first part of the lemma follows for any Kauffman state of . dn ∗ 10 with at least one B smoothing. We find that its deformed curve . h ' intersects any Kauffman state of . dn ∗ 10 at most 2n times. Thus, any curves .(qi, pi ) = qpii satisfy . pi ≤ n (in fact . |pi | ≤ n). See Figs. 14.7a and b. A similar proof works for the denominator: if a Kauffman state has at least one A smoothing and one B smoothing, then we can always find them to be neighbors (along a circle) and draw a . 10 curve which intersects a diagram of . dn ∪ 10 smoothed at those A and B crossings in no more than .2(d − 1) points. Compare Figs. 14.7 and 14.8. This can be written more generally by first observing that the A and B crossings do not have to be neighbors. Thus, we can pair A and B smoothings until there are no A or B smoothings to be paired. Thus, consider the Kauffman state with .n A A smoothings and .n B = 2d − n A B smoothings. Let .nc be the minimum of these numbers. Then the corresponding .qi satisfy .qi ≤ d − nc . u n 5 In this proof we use notation from the book [PH]; compare Fig. 14.6.

14.5 The Algorithm

279

(1,1) parameter

(2,2) parameter

(1,−2) parameter

(2,1) parameter

(5,0) parameter

(5,3) parameter

Fig. 14.6: Illustrations of Dehn coordinates

280

14 Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened

xi−1 xi

xi

(a) Passing via B-crossing.

(b) Passing via neighboring A and B crossings.

Fig. 14.7: Illustration of . h '

Fig. 14.8: Examples of A-B passing

Corollary 14.5.8 Consider the product of (not necessarily reduced) positive curves n1 n2 d1 ∗ d2 with . d = d1 n2 − d2 n1 > 0, then in the skein module the product can be written as: (d1, n1 ) ∗ (d2, n2 ) = A2d (d1 + d2, n1 + n2 ) + A−2d (d1 − d2, n1 − n2 ) E . + ui · (qi, pi ),

.

i

where .0 ≤ pi < n1 + n2 and .0 ≤ qi < d1 + d2 . Proof Assume that the positive curves .(d1, n1 ), (d2, n2 ) are obtained from the pair (d, n) ∗ (0, 1) by the action of a nonnegative matrix of .SL(2, Z) (upper half of the Farey diagram illustrated in Fig. 14.4). In other words, there exist .n1 and .d1 so that:

.

[ .

n2 a1,2 d2 a2,2

][

] [ ] n1 n n = 1 2 . d1 d2 d0

The condition (coming from Theorem 14.5.7)6 is now as follows: .0 ≤ pi ≤ n1 − a1,2 , 0 ≤ qi ≤ d1 − a2,2 . u n

.

The following result can be thought as a stronger version of part (2) of Lemma 14.5.4. E 6 Let .(d, n)∗(0, 1) = A2d (d, n+1)+ A−2d (d, n−1)+ i ui ·(qi' , pi' ). According to Theorem 14.5.7, ' ' ' ' .0 ≤ pi ≤ n and .0 ≤ qi ≤ d − 1. Therefore, . pi = n2 pi + a1,2 qi ≤ n2 n + a1,2 (d − 1) = n1 − a1,2 and .qi = d2 pi' + a2,2 qi' ≤ d2 n + a2,2 (d − 1) = d1 − a2,2 .

14.6 The Positivity Conjecture

281

Corollary 14.5.9 Consider the curves .(d1, n1 ) and .(d2, n2 ) where .d2 ≥ 0 and consider the product E 2d −2d2 .(d2, n2 ) ∗ (0, 1) = A 2 (d2, n2 + 1) + A (d2, n2 − 1) + wi · (qi, pi ). i

If . |n2 d1 − n1 d2 | = 1, then .

|n1 qi − d1 pi | ≤ d1 .

Proof By Theorem 14.5.7, we have .0 ≤ qi ≤ d2 − 1. From Lemma 14.5.3 (2) and Lemma 14.5.4 (1), we have .

|n1 qi − d1 pi | ≤

d1 qi |n2 d1 − n1 d2 | + |n2 qi − d2 pi | < 1 + d1 . d2 d2

Thus, .

|n1 qi − d1 pi | ≤ d1,

as needed. u n

14.6 The Positivity Conjecture The algorithm and data presented in this lecture suggest that the Chebyshev basis for S alg (F0,4 ) in [BMPSW] is positive. Moreover, the authors of [BMPSW] conjectured the following:

.

Conjecture 14.6.1 ([BMPSW]) E (1) If .(d2, n2 )T ∗(d1, n1 )T = i wi ·(qi, pi )T , where .wi ∈ K0, then .wi is a nonnegative polynomial in the variables . A, R0,1, R1,1, R1,0 , and y. n1 d2 −n2 d1 (d + d , n + n ) + A−n1 d2 +n2 d1 (d − d , n − (2) If .(d2, nE 2 )T ∗ (d1, n1 )T = A 1 2 1 2 T 1 2 1 n2 )T + i vi · (qi, pi )S , with .vi ∈ K0 , then .vi are nonnegative polynomials in the variables . A, R0,1, R1,1, R1,0 , and y.

All the computations and the formulas in Theorems 14.4.6 and 14.4.7 support Conjecture 14.6.1. The last couple of years have seen a flurry of activity in proving this conjecture. Lower and upper bounds for the positive bases of skein algebras were discovered in [LTY]. In particular, it was shown that any sequence .(Pn ) of polynomials which forms a positive basis for the KBSA of a thickened surface, with genus -n ) ≤ (Pn ) ≤ (Sn ). .g ≥ 1 or with at least four punctures, satisfies the inequality .(T ±1 Here, .(Pn ) ≥ (Q n ) implies that each polynomial .Pn is a .Z ≥0 [A ]-linear combination of the polynomials .Q0, Q1, . . . , Q n .

282

14 Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened

In [Bou], Bousseau resolved Conjecture 14.6.1 in the affirmative and proved positivity for the structure constants of the bases of .S alg (F0,4 ) described in the conjecture. He also resolved the positivity conjecture for .S alg (F1,1 ). The key idea of the proof is the realization of the skein algebra of the four-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov-Witten theory applied to a complex cubic surface. Thus, the bases of the KBSA of .F0,4 described in [BMPSW] are positive and are sandwiched between the two types of Chebyshev polynomials described in [LTY]. We end this lecture with an open question about the computational complexity of the algorithm discussed in [BMPSW]. Open Question 1. What is the time complexity of multiplying fractions in the .S alg (F0,4 )? 2. Is the complexity of the algorithm quicker than exponential, for example, quasipolynomial in terms of the determinant of the curves being multiplied (i.e., of complexity .(det)log(det) where .det is the determinant of a pair of fractions)?

Lecture 15 Spin Structure and the Framing Skein Module of Links in 3-Manifolds

In this lecture, we show that the only way of changing the framing of a knot or a link by ambient isotopy in an oriented 3-manifold is when the manifold has a properly embedded non-separating sphere. This change of framing is given by the Dirac trick, also known as the light bulb trick. We begin with background information on non-separating spheres, Dehn homeomorphisms, and mapping class groups of 3-manifolds. The main tools used in the proof are based on Darryl McCullough’s work on the mapping class groups of 3-manifolds, and spin structures. We then relate these results to the theory of skein modules.

15.1 Introduction Following [BIMPW] , we show that the only way of changing the framing of a knot or a link by ambient isotopy in an oriented 3-manifold is when the manifold has a properly embedded non-separating sphere. As mentioned in Appendix B, framed knots are annuli embedded in a 3-manifold up to ambient isotopy. These ribbons have a core circle which can be knotted. If M is an integral homology sphere (respectively, rational homology sphere), then every knot in M has a preferred framing (respectively, rational framing). In fact, in Sect. 15.5 we show that for arbitrary oriented 3-manifolds we can only define a modulo 2 framing which reflects the affine space of spin structures over .H 1 (M, Z2 ). If M can be embedded in a rational homology sphere, then K has a preferred rational framing (depending on the embedding). In particular, the framing cannot be changed by an ambient isotopy of M. Notice that this does not apply if M has a properly embedded non-separating closed surface, since then it cannot be properly embedded in any rational homology sphere. In this lecture we will show that the only way to change the framing of a knot or link via ambient isotopy is when the manifold M has a non-separating sphere. This © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_15

283

284

15 Spin Structure and the Framing Skein Module of Links in 3-Manifolds

framing change is accomplished via the light bulb trick pictured in Figs. 15.2 and 15.3. The framing change relates to the fact that .π1 (SO(3))  Z2 . Further, because of the invariance of spin structure (Sect. 15.5), the framing can only change by an even number of full twists. An alternative argument can also be made using handle sliding. The lecture starts with a brief history of the problem, after which we turn to some preliminary results from elementary topology. In Sect. 15.4, we give a proof of the main result. Finally, we consider skein modules. The framing change results can be generalized to skein modules via spin structure.

15.2 History In the fall of 1989, the first author was visiting Michigan State University and working on the Kauffman bracket skein module of lens spaces. It was here that he first considered the possibility of changing framings of knots in 3-manifolds via ambient isotopy and realized that such changes are impossible for irreducible 3manifolds. Following Darryl McCullough’s work, Yongwu Rong observed that if M is a compact 3-manifold, then the framing of a knot in M can be changed if and only if M has a non-separating .S 2 [HP2]. The work in [McCu1] and [MM] indicated that the Dirac trick was the only way to accomplish a framing change. The first author then stated, but never proved, Theorem 15.4.1 several times, inspiring Vladimir Chernov to write his own proof in [Che]. In this paper Chernov wrote, This result (for compact manifolds) was first stated by Hoste and Przytycki [HP2]. They referred to the work [McCu1] of McCullough on mapping class groups of 3manifolds for the idea of the proof of this fact. However to the best of our knowledge the proof was not given in the literature. The proof we provide is based on the ideas and methods different from the ones Hoste and the first author had in mind. The proof of Theorem 15.4.1 found in [BIMPW] employs results from McCullough about the mapping class group of 3-manifolds. See Appendix B for a description of the classical mapping class group of surfaces. In [McCu2], McCullough uses Papakyriakopoulos’ loop theorem [Pap2] and its generalization to the annulus proven by Friedhelm Waldhausen [Wal1]. The application of spin structure to skein modules was considered by John W. Barrett and Vladimir Turaev. As in their work, we view spin structures as parallelizations of the tangent bundles of a 3-manifold.1

1 After [BIMPW] was put up on arXiv, the authors were informed by Chernov about his new paper [CCS] where they prove the result for knots using McCullough’s results. Thus, the novelty of [BIMPW] is the result for links and its relation to skein modules.

15.3 Preliminaries

285

15.3 Preliminaries In Appendices A and B, we introduce the following preliminary material from 3-dimensional topology: incompressible surfaces, mapping class groups of 3manifolds, and Dehn homeomorphisms. In this section, we provide relevant definitions and results pertinent to the proof of Theorem 15.4.1. Definition 15.3.1 A non-separating sphere is a sphere embedded into M such that M \ S 2 is connected.

.

Definition 15.3.2 Let M be an orientable manifold and .Homeo(M) denote the space of PL orientation preserving homeomorphisms of M. Then the mapping class group of M, denoted by .H (M), is defined as the space of all ambient isotopy classes of .Homeo(M). Definition 15.3.3 Let .(F n−1 × I, ∂F n−1 × I) ⊂ (M n, ∂ M n ), where F is a connected codimension 1 submanifold, and .(F × I) ∩ ∂ M = ∂F × I. Let . φt  be an element of . π1 (Homeo(F), 1F ), that is, for .0 ≤ t ≤ 1, . φt is a continuous family of homeomorphisms of F such that .φ0 = φ1 = 1F . Define a Dehn homeomorphism as . h ∈ H (M) by:  .

h=

h(x, t) = (φt (x), t) h(m) = m

if (x, t) ∈ F × I if m  F × I.

Dehn homeomorphisms are generalizations of Dehn twist homeomorphisms of a surface. When .π1 (Homeo(F)) is trivial, a Dehn homeomorphism must be isotopic to the identity. Define the Dehn subgroup .D(M) of .H (M) to be the subgroup generated by all Dehn homeomorphisms. The following table lists .π1 (Homeo(F)) for connected surfaces and the names of the corresponding Dehn homeomorphisms of oriented 3-manifolds. F × 1×I S . D2 S2 X(F) < 0 S1

S1

π1 (Homeo(F))

Dehn homeomorphism

Z×Z Z Z Z/2Z {0}

Dehn twist about a torus Dehn twist about an annulus Dehn twist about a disk rotation about a sphere

Theorem 15.3.4 (Finite Mapping Class Group Theorem [HM]) Let M be a closed oriented irreducible non-Haken 3-manifold, that is, M is irreducible and contains no properly embedded, incompressible, two-sided surface. Then .H (M) is finite. This theorem is also true for all hyperbolic 3-manifolds.

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15 Spin Structure and the Framing Skein Module of Links in 3-Manifolds

Theorem 15.3.5 ([GMT]) The mapping class group of a compact hyperbolic 3manifold is finite. When the authors of [BIMPW] decided to give details of the old proof, they noted that from McCullough’s new paper, [McCu1], the proof could be deduced in a relatively simple way (but still using the generalized loop theorem for the annulus). Soon after, the authors found another paper by McCullough [McCu2] from which the main result follows more directly. This result and a corollary are stated below. Theorem 15.3.6 ([McCu2]) Let M be a compact, orientable, 3-manifold, and let h be a homeomorphism which is Dehn twists on the boundary about a collection of simple closed curves: .C1, C2, ...Cn . Then there is a collection of disjoint compressing disks and incompressible annuli such that each boundary circle of this collection is isotopic to one of the .Ci . Further h is isotopic to a composition of Dehn twists about these disks and annuli on .∂ M. Therefore, h is the result of a composition of Dehn twists about a collection of disjoint annuli and disks with a homeomorphism that is the identity on the boundary. The next corollary follows directly from this theorem. Corollary 15.3.7 Let . M  be a compact oriented 3-manifold with some boundary components, say, .∂1 (M ), . . . , ∂k (M ) being tori. Let .  f : M  −→ M  be a home . ∂i (M ) and which is the identity on omorphism which acts nontrivially on every   ∂ (M  ). Then either .∂ (M  ) has a compressing disk, say . D2 , or there is .∂ M \ i i i i some . j  i such that there is an incompressible annulus .Anni, j with one boundary component on .∂i (M ) and the other on .∂j (M ). Furthermore, one can take these disks and annuli to be disjoint. Also, .  f restricted to .∂ M  is ambient isotopic to the composition (of some powers) of the Dehn homeomorphism along . Di2 and .Anni, j . Exercise 15.3.8 Verify Corollary 15.3.7 by arguing that the two boundary disks must lie on separate tori. For the remainder of the lecture we will denote a framed knot (respectively link) by .K (respectively .L ) and the underlying unframed knot (respectively link) by K (respectively L).

15.4 Main Result We give a formal statement and proof of the main theorem for this lecture. Theorem 15.4.1 ([BIMPW]) Let .L be a framed link in a compact oriented 3manifold M. The only way of changing the framing of .L by ambient isotopy while preserving the components of L is when M has a properly embedded non-separating 2-sphere and either:

15.4 Main Result

287

Ann

Fig. 15.1: A homeomorphism f satisfying the premise of Theorem 15.4.1 gives rise to a collection of disjoint compressing disks and annuli. This picture features a framed link .L =   . There are two non-separating spheres in M in this picture

(i) L intersects the non-separating 2-sphere transversely exactly once, or (ii) L intersects the non-separating 2-sphere transversely in two points, each point belonging to a different component of L. The framing is changed by a composition of even powers of the Dehn homeomorphisms along the disjoint union of .S 2j , where .S 2j satisfy conditions (i) or (ii). We illustrate the change of framing by the light bulb trick (in the regular neighborhood of .S 2 ) in Fig. 15.2 if there is one point of intersection and in Fig. 15.3 if there are exactly two points of intersection of L with .S 2 . Note that the only way an oriented 3-manifold can have a non-separating 2-sphere is if it contains a copy of .S 1 × S 2 in its prime decomposition. Proof We follow the proof in [BIMPW]. Let .L be a framed link in a compact oriented 3-manifold M with components .K 1 , .K 2, ..., K n . Suppose . f : M → M is a homeomorphism that is ambient isotopic to the identity, changes the framing of .L , and preserves the boundary of M. By [Hud], we can choose a regular neighborhood .VKi such that . f (VKi ) = VKi . We remove the union of these regular neighborhoods of .L from M to obtain  M  = M \ VKi . . M  has a collection of toroidal boundary components: .T1, T2, ...Tn . Let . f˜ = f  M  . Then . f˜ satisfies the premise of Theorem 15.3.6. Since f fixes .∂ M, . f˜ can only change .∂ M  on the boundary tori. Suppose .Ti is a boundary tori changed by . f˜. By Corollary 15.3.7, either .Ti has a compressing disk, say . Di2 , or there is some . j  i such that there is an incompressible annulus .Anni, j with one boundary component on .Ti and the other on .Tj . These disks and annuli are disjoint and form the building blocks for the non-separating spheres. See Fig. 15.1 for an illustration.

.

Let us first consider the case of the compressing disk. The original homeomorphism, f , is defined on all of M. To be able to extend the homeomorphism . f˜ defined on . M  to the entire manifold, the boundary of the compressing disk must be the same as the boundary of the meridian disk, . Dμi , of the tubular neighborhood. Gluing the compressing disk to this meridian disk results in a non-separating .S 2 (see Fig. 15.1).

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15 Spin Structure and the Framing Skein Module of Links in 3-Manifolds

This sphere intersects L transversely in exactly one point since the meridian disk of the tubular neighborhood is intersected by .Ki once. Now consider the annular case. .Ci and .C j must lie on different boundary tori. Again, to be able to extend the homeomorphism, . f˜ to all of M, the boundary circles of the incompressible annulus must be the boundaries of meridian disks . Dμi and . Dμ j . Gluing these disks to the corresponding boundary circles of the annulus results in a non-separating .S 2 . The link, L, intersects this sphere in precisely two places: where it would intersect the meridian disks, . Dμi and . Dμ j . Let .S12, S22, ..., Sn2 denote the collection of two spheres constructed above. We must show that the framing is changed by a composition of even powers of the Dehn homeomorphisms along the disjoint union of these spheres. Let .τi denote the Dehn homeomorphism along the sphere .Si2 . .τi2 is isotopic to the identity map since . π1 (SO(3))  Z2 . The function f is not only a homeomorphism, but it is ambient isotopic to the identity map and therefore also to .τi2 . In the next section, we introduce spin structures on oriented 3-manifolds. Their existence guarantees that f can be considered as an even number of rotations .τi2 about the collection of non-separating spheres . 

Isotopy

Isotopy

Isotopy

Isotopy

Isotopy

(2)

Fig. 15.2: Dirac trick for a knot illustrated using a light bulb In the next section, we investigate spin structures, which form a bridge between the result above and various types of skein modules. Spin structure also allows us to prove that we must use an even number of Dehn twists and that the framing can only change by an even value.

15.5 Spin Structures

289

Isotopy

Isotopy

(2) Isotopy

Isotopy

Isotopy (−2)

Fig. 15.3: Dirac trick for a link with two components illustrated using a light bulb. For a nice computer animation of the Dirac trick we refer the reader to [DFHHHKPS]

15.5 Spin Structures Barrett was the first to connect spin structure to skein modules in [Bar]. In this work he proved that if there exists a spin structure, s, on an oriented 3-manifold, M, then for any knot K in M we can create a map, .spin : S2,∞ (M, R, A) → Z2 such that the following theorem holds. Theorem 15.5.1 ([Bar]) Each spin structure, s, for M determines an isomorphism φ(s) : S2,∞ (M, R,A) −→ S2,∞ (M, R, −A). If L is a link with components .{γ1, ..., γn }, n then .φ(L) = (−1) i=1 spin(γi ,s) L.

.

As discussed in Lecture 13, the R-module isomorphism described in Theorem 15.5.1 can be extended to an isomorphism of R-algebras [PS3]. The existence of the .spin map used in Theorem 15.5.1 relies on the following. Definition 15.5.2 M is parallelizable if the tangent bundle of M is trivial, that is, there are three vector fields .V1 , .V2 , and .V3 which form a basis at every tangent space. Theorem 15.5.3 ([Sti]) Every orientable 3-manifold is parallelizable.

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15 Spin Structure and the Framing Skein Module of Links in 3-Manifolds

In our proof of Theorem 15.4.1, we still need to show why the framing of a knot K in M cannot be changed by one full twist using ambient isotopy. The result follows from the following application of spin structure. In the following discussion, let .TM be the tangent bundle for a 3-manifold, M, and .V : M −→ TM a vector field. Assume that V is nondegenerate. Let M be an oriented 3-manifold.

.

We view the spin structure as a homotopy class of parallelizations. After choosing a parallelization, we can identify the spin structure with the space .H 1 (M, Z2 ). By the universal coefficient theorem, we have that .H 1 (M, Z2 )  Hom(H1 (M), Z2 ). Each knot .K ⊂ M represents an element of .H1 (M, Z2 ). We define a map that takes this .K to an element of .Z2 . This map will be a homomorphism between .H1 (M, Z2 ) and .Z2 .

Fig. 15.4: The .gi form a closed loop. This leads to a homomorphism from . H1 (M, Z2 ) to .Z2 Fix a point .ma ∈ M. Let .(V1, V2, V3 ) be an orthonormal parallelization of M. Let K be a framed knot in M. Then .K ∈ H1 (M, Z2 ). To our point .ma we affix the vectors .v1, v2, v3 . Fix another point .mk ∈ K . We can affix the three orthonormal vectors .(VT , Vfr, V3 ) to this point. See Fig. 15.4 for details. There is an element .g ∈ SO(3) that maps .(v1, v2, v3 ) to .(vT , vfr, v3 ). Traveling along the knot we produce a closed loop of elements in .SO(3). Thus, we have an element of .π1 (SO(3)) which is isomorphic to .Z2 . Concatenating the map we have constructed with this isomorphism, we obtain a map from .H1 (M, Z2 ) to .Z2 . So using the parallelization we can build a homomorphism: . φ: . H1 (M, Z2 ) → Z2 . .

Proposition 15.5.4 ([BIMPW]) The framing of a knot can only be changed via ambient isotopy by an even number of twists.

15.6 Applications to Skein Modules

291

K ) in .Z2 . This is because Proof The map .φ sends .K (1) to the opposite element of .φ(K the frame .(vT , vfr, v3 ) must go around the knot one more time to end in the same K (n+1) ) = φ(K K (n) ) + 1. position. So we have .φ(K By way of contradiction, suppose .γ is an ambient isotopy that changes the framing K (n) ) = K (n+1) . Then, we know the composition of .K by one twist. That is, suppose .γ(K of .φ with .γ is a homomorphism from the first homology group of M into .Z2 . So, .[φ ◦ γ] ∈ Hom(H1 (M), Z2 ). Also, since the homomorphism group is equivalent to the first cohomology group, we know that .[φ ◦ γ] ≡ [φ]. However, .φ ◦ γ sends .K (n) to the opposite element of .K (n+1) in .Z2 . That is, K ) = φ(K K (n+1) ) = φ(K K (n) ) + 1  φ(K K (n) ). This is a contradiction. Hence, such a .(K  map .γ cannot exist. (n)

In the next section, we use Theorem 15.4.1 and spin structure to prove new results for various skein modules. Remark 15.5.5 For oriented 3-manifolds M for which .H (M ) is finite, Theorem 15.4.1 can be deduced immediately. This is because . f |∂VK i has infinite order in .H (∂VKi ), and thus, .  f has infinite order in .H (M ). Hence, we get a contradiction. This happens, in particular, when . M  is a hyperbolic manifold or a closed oriented irreducible non-Haken 3-manifold (see Theorems 15.3.4 and 15.3.5).

15.6 Applications to Skein Modules In Lectures 11 and 12, we define the framing skein module, .S0 (M; q); q-homology → − fr

skein module, .S2 (M; q); and Kauffman bracket skein module, .S2,∞ (M). The following is equivalent to Theorem 15.4.1. Theorem 15.6.1 ([BIMPW]) For an oriented 3-manifold M, S0 (M; q) = Z[q±1 ]L f ⊕



.

L ∈(L fr \L f )

Z[q]

, q2 − 1

where .L f is composed of links which do not intersect any 2-sphere in M transversely at exactly one point. Proof By Theorem 15.4.1, we have two possible ways to change the framing of .L . Case 1: Suppose L intersects a non-separating .S 2 transversely in exactly one point. That is, suppose .L  L f . Let .K i be the intersecting component. Then by Theorem 15.4.1, the Dehn homeomorphism .τi2 (which is a Dehn twist along the sphere .S 2 ) is ambient isotopic to the identity and twists the framing of .K i by two full twists.

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15 Spin Structure and the Framing Skein Module of Links in 3-Manifolds

Observe that in .S0 (M; q) we have the following: .

K K (1) i − qK i ≡ 0 2 ⇐⇒ K (2) i ≡ q Ki

K i ) ≡ q2K i ⇐⇒ τi2 (K ⇐⇒ K i ≡ q2 K i K i ≡ 0. ⇐⇒ (1 − q2 )K L = 0. When we Taking this relation for every component of .L , we have .(q2 − 1)L Z[q]  L we obtain .Z[q±1 ]L f ⊕ . quotient .Z[q±1 ]L fr by .(1 − q2 )L 2 L ∈(L fr \L f ) q − 1 Case 2: We now suppose .L ∈ L f . So if the framing of .L changes, then there is a non-separating .Si,2 j and two components .Ki and .K j that intersect 2 .S i, j transversely in one point each. Then .τi changes the framing of .K i by two positive full twists and changes that of .K j by two negative full twists. In .S0 (M; q) the twists cancel algebraically since the framing skein module cannot see which component is twisted. In this case, under the quotient relation, ±1 ]L f . .L is unchanged and mapped into .Z[q  Corollary 15.6.2 ([BIMPW]) There exists an epimorphism .ω from the framing skein module to the .Z[q]/(q2 − 1)-group ring over the first homology with .Z2 coefficients, that is,

Z[q] H1 (M, Z2 ), .ω : S0 (M; q) −→ q2 − 1 which is not canonical and depends on the choice of spin structure (in the form of a parallelization of a tangent bundle to M). The choice of parallelization gives a map .b : H1 (M, Z2 ) −→ Z2 . Hence, .ω(K) = |K | if .b(|K |) = 0 and .ω(K) = q|K | if . b(|K |) = 1. Here .K is a framed knot in M and . |K | denotes the homology class of K in .H1 (M, Z2 ). Similar results hold for the Kauffman bracket skein module and the q-homology skein module. Using an argument analogous to that in Theorem 15.5.1, there is ±1 ] a map .φ˜ : Z[A±1 ]L fr → AZ[A 4 +A2 +1 . We have the following observations regarding . H1 (M, Z2 ) and the Kauffman bracket skein module. Proposition 15.6.3 ([BIMPW]) There exists an epimorphism

Z[A] H1 (M, Z2 ), . φ : S2,∞ (M) −→ A4 + A2 + 1 which is not canonical and depends on the choice of spin structure on M (here in the form of a parallelization of a tangent bundle to M). The codomain of .φ is called spin twisted homology. Compare with Corollary 15.6.2 and [Bar, PaSa].

15.7 Exercises

293

Exercise 15.6.4 Prove Proposition 15.6.3 by checking that the map .φ˜ satisfies the skein relation and the initial condition. In [Prz12], it is proven (in the context of q-homology skein modules) that if M has no non-separating closed surface F (that is, F is properly embedded in M and . M \ F has the same number of components as M), then one cannot change the framing of a knot by ambient isotopy. In the q-homology skein module, we have the following result analogous to Theorem 15.5.1. Proposition 15.6.5 ([BIMPW]) There exists an epimorphism

→ − Z[q] fr H1 (M, Z2 ), .ψ : S (M; q) −→ 2 q2 − 1 which is not canonical and depends on the choice of spin structure on M. As before, compare with Corollary 15.6.2. Exercise 15.6.6 Prove Proposition 15.6.5. Start by arguing that there is a map .b : L ) = |L L | is .b(|L L |) = 0 and .ψ(L L ) = q|L L | if .b(|L L |) = 1. H1 (M, Z2 ) → Z2 such that .ψ(L Next check that the map .ψ satisfies the skein relations of q-homology. The following diagram summarizes the results of Corollary 15.6.2 and Propositions 15.6.3 and 15.6.5.

15.7 Exercises Exercise 15.7.1 Show that for the homology sphere there is a preferred framing for every knot.

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15 Spin Structure and the Framing Skein Module of Links in 3-Manifolds

Exercise 15.7.2 Let M be a rational homology sphere, which is a closed three dimensional manifold with the same homology over Q as S 3 . Consider a knot K in M. Let VK be the tubular neighborhood of the knot, and let M  = M \ int(VK ). Show that there is a simple closed curve on ∂VK that bounds an oriented surface in M . Exercise 15.7.3 Using the previous two exercises, prove the main result of this lecture – Theorem 15.4.1 – for rational homology spheres.

Lecture 16 The Witten-Reshetikhin-Turaev Invariants of 3-Manifolds

The Witten-Reshetikhin-Turaev invariants are fascinating 3-manifold invariants with connections to physics. In fact, their existence was predicted by Edward Witten in his work on Chern-Simons gauge theory and topological quantum field theory. The invariants also have close connections to skein theory. In this lecture, we focus on W. B. Raymond Lickorish’s Kauffman bracket skein theoretic approach to the invariants. In the last section of this lecture, we introduce a skein module described by Justin Roberts and Adam S. Sikora that incorporates the Jones-Wenzl idempotents and has close connections to the .(2 + 1)-TQFT and WRT invariants.

16.1 Introduction In this lecture, we will introduce the colored Jones polynomials and the WittenReshetikhin-Turaev invariants. Then we will provide methods to computing the Witten-Reshetikhin-Turaev invariants via Gauss sums. In the last section, we will focus on a skein module introduced by Justin D. Roberts and Adam S. Sikora in [Rob1, Sik2] which we call the Witten skein module, named after Edward Witten due to his vision of a connection between topological quantum field theory (TQFT) and the Jones polynomial.

16.2 The Witten-Reshetikhin-Turaev Invariant The 3-manifold invariants given in this section were introduced by Edward Witten using Chern-Simons gauge theory and defined using the axioms of TQFT [Wit]. The existence of the 3-manifold invariants was proven by Nicolai Y. Reshetikhin and Vladimir G. Turaev in [RT] using modular Hopf algebras associated with quantum © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_16

295

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16 The Witten-Reshetikhin-Turaev Invariants of 3-Manifolds

groups. A year later, Kevin Walker proved that Reshetikhin and Turaev’s definition is equivalent to the definition that uses the TQFT axioms [Walk]. W. B. Raymond Lickorish in [Lic6, Lic5, Lic7] constructed 3-manifold invariants for 4rth roots of unity, where .r ≥ 3, by using the Kauffman bracket skein module and the TemperleyLieb algebra. Later, Christian Blanchet, Nathan Habegger, Gregor Masbaum, and Pierre Vogel in [BHMV1] extended Lickorish’s construction to include 2pth roots of unity for p odd by using cablings instead of the Temperley-Lieb algebra. Then in [BHMV2], the authors produced a topological construction of the TQFTs associated with a renormalized version of the 3-manifold invariants they had constructed in their previous paper [BHMV1]. There are four approaches to the Witten-Reshetikhin-Turaev invariant: combinatorial, TQFT axioms, Feynman path integrals, and skein theory. We will focus on the skein theoretic approach given by Lickorish in [Lic8], also known as the .SU(2) Witten-Reshetikhin-Turaev invariant. This definition heavily relies on the Kauffman K × I; C(A)). A definition bracket skein module of the annulus cross an interval, .S2,∞ (K may be found in Lecture 12. This lecture also relies on knowledge gained in Lecture 9, particularly the definition of the Jones-Wenzl idempotent . fn , and the notations n A2n+2 −A−2n−2 . .tr Ann ( fn ) = Sn (z) and .Δn = (−1) A2 −A−2 We start by defining a multilinear map where we illustrate how to decorate (also known as threading) each component before evaluating the decorated link in the Kauffman bracket skein module of .S 3 . Then we use these tools to define the colored Jones polynomial before defining an element .Ω such that the decoration of a link with .Ω results in invariance under the Kirby 2 move. Finally, we explain the normalization needed in order to obtain invariance under the Kirby 1 move and the renormalization needed to compare it to Witten’s definition. See Appendix B for the definition of Kirby moves. Definition 16.2.1 Let .L be a framed link with k components in .S 3 , and order the k . Then the meta-bracket, . > , is a K i }i=1 components such that .L = {K L multilinear map associated with .L , .

>L : K 1 × I; C(A)) × · · · × S2,∞ (K K k × I; C(A)) → S2,∞ (S 3 ; C(A)). S2,∞ (K

It is enough to define the meta-bracket on the basis; we will describe this diagramk as a diagram using blackboard framing. For each K i }i=1 matically, by viewing .L = {K h .1 ≤ i ≤ k, if the ith component is equal to . z for . h > 0, then replace .Ki with h parallel copies, and if the ith component is equal to 1, then remove .Ki from the diagram. Finally, the image is the unreduced Kauffman bracket polynomial of the resulting link. Example 16.2.2 We will calculate . L for .L given in Fig. 16.1. First use the linearity of the map, . L = AL + AL .

16.2 The Witten-Reshetikhin-Turaev Invariant

297

Fig. 16.1: Illustration of decorating links

Since

and

,

where .d = −A2 − A−2 , then . L = A(−A2 − A−2 )3 + A(−A2 − A−2 )4 . Recall from Corollary 9.4.2 that the trace of the nth Jones-Wenzl idempotent is equal to the nth Chebyshev polynomial of the second kind in the variable z, .tr Ann ( fn ) = Sn (z). This trace can be viewed as an element of the Kauffman bracket skein module .S2,∞ (Ann × I). For example,

.

We can immediately obtain a version of the colored Jones polynomial from the meta-bracket and the trace of the Jones-Wenzl idempotent. Informally, the nth colored Jones polynomial of a knot, up to a factor, is obtained by considering a diagram of a zero-framed knot in blackboard framing, decorating the knot with the .(n − 1)th Jones-Wenzl idempotent, and then evaluating the knot by the unreduced Kauffman bracket polynomial. Definition 16.2.3 The nth colored Jones polynomial1 of a zero-framed knot .K , denoted by . JK (n), is defined as .

JK (n) = K .

The Jones polynomial can be obtained from the colored Jones when .n = 2. In particular, . JK (2) = (−t 1/2 − t −1/2 )VK (t) , where . A = t −1/4 and .VK (t) is the Jones polynomial of the underlying unframed knot K of .K . This can be defined similarly for zero-framed links.

1 There are many different versions of the colored Jones polynomial due to various different conventions and normalizations. For example, sometimes, it is advantageous to define the colored Jones polynomial as .JK (n) = (−1) n−1 < >K with the extra .(−1) n−1 factor added so that 2n −2n . See [Le1, LeTr]. when K is the unknot, .JK (n) = AA2 −− A A−2

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16 The Witten-Reshetikhin-Turaev Invariants of 3-Manifolds

Fig. 16.2: An unknot decorated by .Ω embedded in .Ann × I along with an arc joining two boundary points. a denotes the curve fixed on the boundary that is above .Ω, and b denotes the curve fixed on the boundary that is below .Ω

The colored Jones polynomial is not a 3-manifold invariant; however, the WittenReshetikhin-Turaev 3-manifold invariant that Lickorish in [Lic8] constructs uses the same tools. We will now show this construction by first presenting a linear combination of the trace of Jones-Wenzl idempotents that yield invariance under a Kirby 2 move. Definition 16.2.4 Let A be a primitive 4rth root of unity, .r ≥ 3 be an integer, and .L be a framed link in .S 3 . Let .Ω ∈ S2,∞ (Ann × I) be defined by Ω=

r−2 E

.

Δn trAnn ( fn )

n=0

and .L Ω ∈ S2,∞ (S 3 ) be defined by .L Ω = L , where .L Ω is calculated by decorating the components of .L by .Ω and then evaluating it in .S2,∞ (S 3 ). Then .L Ω is an invariant under a Kirby 2 move of . ML , where . ML is a closed orientable 3-manifold obtained by surgery from .L ∈ S 3 . A proof can be found in [Lic8] and heavily relies on Lemma 16.2.5. Lemma 16.2.5 ([Lic8]) In .S2,∞ (Ann × I, 2 points.), .aΩ − bΩ is a linear sum of two elements, each of which contains a copy of . fr−1 . More precisely,

.

(16.1)

Proof We use Hans Wenzl’s theorem, Theorem 1.20 in Lecture 9, where we close the bottom .n − 1 strands in the annulus and place the remaining strands fixed to the boundary of the annulus as shown in Eq. 16.2.

16.2 The Witten-Reshetikhin-Turaev Invariant

=

299

−

Δ −1 Δ (16.2)

.

We use the idempotent property of . fn and rearrange Eq. 16.2 to obtain Eq. 16.3. −1

Δ

+Δ

−1

=Δ (16.3)

.

By using the notation shown in Fig. 16.2, we have that the right-hand side of Eq. 16.3 is equal to .aΔn Sn (z). We now sum these equalities from .n = 0 to .n = r − 2 r−2 E with the convention that .Δ−1 = 0. The right-hand side is equal to . aΔn Sn (z) = aΩ. n=0

After rotating each annulus by .π radians and taking the sum again, the right-hand side becomes .bΩ. The difference between the left-hand side of the two equations is precisely the right-hand side of Eq. 16.1, that is, the difference of the first term of o Eq. 16.3, when .n = r − 2, and its rotation; each has a copy of . fr−1 . Exercise 16.2.6 Prove that .Δr−1 = 0 when A is a primitive 4rth root of unity. Exercise 16.2.7 Prove that for

we have .L Ω =

−2r . (A2 −A−2 )2

By considering the positive and negative eigenvalues of the linking matrix of .L , denoted by .b+ and .b− , respectively, .L Ω can be normalized to be a 3-manifold invariant. Definition 16.2.8 ([Lic8]) Let A be a primitive 4rth root of unity, .r ≥ 3 be an integer, and .L be a framed link in .S 3 . Let .b+ and .b− denote the number of positive and negative eigenvalues of the linking matrix of .L , respectively. Then .

is an invariant of . ML , where . ML is a closed orientable 3-manifold obtained by integral surgery from .L ∈ S 3 . Theorem 16.2.9 The following properties hold for .φ: 1. .φ(S 3 ) = 1. 2. Let . M1 and . M2 be closed orientable 3-manifolds and . M1 #M2 be its connected sum; then .φ(M1 #M2 ) = φ(M1 )φ(M2 ).

300

16 The Witten-Reshetikhin-Turaev Invariants of 3-Manifolds

Proof The proof of (1) is trivial. Suppose . M1 = ML 1 and . M2 = ML 2 for some framed links .L 1, L 2 ⊂ S 3 ; then . M1 #M2 = ML 1 uLL 2 . The linking matrix has the following property . AL 1 uLL 2 = AL 1 ⊕ AL 2 , and the meta-bracket inherits the multiplicative property . K1 uK2 = K1 K2 from the unreduced Kauffman bracket. Therefore, L 1 u L 2 )Ω −(b L1 )+ −(b L2 )+ −(b L1 )− −(b L2 )− φ(ML1 uL2 ) = (L

.

L 1 )Ω −(b L1 )+ −(b L1 )− (L L 2 )Ω −(b L2 )+ −(b L2 )− = (L = φ(ML 1 )φ(ML 2 ). o Sometimes, it is beneficial to renormalize the invariant in such a way that allows . The result is an invariant dependent on the us to combine signature of the linking matrix of .L . Recall from Appendix B that the signature of a linking matrix is the difference of the number of positive and negative eigenvalues of the link. Lickorish in [Lic8] showed that by solving for an element . μ such that , we obtain A2 − A−2 = .μ = √ −2r

/ ( ) / (π) 2 1 πi/r 2 −πi/r (e sin . −e ) = r 2i r r

Definition 16.2.10 ([Lic8]) Let A be a primitive 4rth root of unity, .r ≥ 3 be an L ) be the signature of the linking matrix of .L , integer, .L be a framed link in .S 3 , .σ(L and . ML be a closed orientable 3-manifold obtained by surgery from .L ∈ S 3 . Let . μ ∈ C(A) be defined by as shown above, and let .L μΩ = L . Then the renormalized invariant is defined by

, then .I(ML ) = μk+1−b+ −b− φ(ML ),

Remark 16.2.11 Since

where k is the number of components of the framed link .L . Notice that the value k + 1 − b+ − b− is invariant under Kirby 1 and 2 moves.

.

Example 16.2.12 . Since

1.

, then

. . From Exercise 16.2.7,

2. , this implies that , then

. Since we also have

16.3 Calculations of the Witten-Reshetikhin-Turaev Invariant and Gauss Sums

301

The connected sum property of .I follows from Theorem 16.2.9. Theorem 16.2.13 Let . M1 and . M2 be closed orientable 3-manifolds and . M1 #M2 be its connected sum; then .I(M1 #M2 )I(S 3 ) = I(M1 )I(M2 ). Remark 16.2.14 For the ambitious reader. 1. Renormalization given in Definition 16.2.10 is necessary to obtain the 3manifold invariant . Z(M) for the Lie (gauge) group .G = SU(2) and . k = r − 2 described by Witten in [Wit] . For example, the invariant . Z(M) satisfies / ( π ) 2 3 and Z(X × S 1 ) = dim(HX ), sin . Z(S ) = k +2 k +2 where . X = Fg,0 is a surface of genus g and .HX is the Hilbert space of X. The relation described on page 137 in [Wit] also holds, .

Z(M1 #M2 )Z(S 3 ) = Z(M1 )Z(M2 ).

2. It was proven in [GW] that it is necessary to equip the manifold with extra structure2 in order to obtain a TQFT from the 3-manifold invariant.

16.3 Calculations of the Witten-Reshetikhin-Turaev Invariant and Gauss Sums Carl Friedrich Gauss introduced the Gauss sum (also known as a quadratic Gauss E 2π i mn 2 k , in July 1801 in [Gau1]. Then in [Gau2], he presented an evalusum), . k−1 n=0 e ation of the Gauss sum for all positive integers k. Gauss initially worked on a special case where .m = 1 and k is odd, as seen on√page 438 √ of [Gau3]. In this case, he was able to show that the sum evaluates to .± k or .±i k, for . k ≡ 1 mod 4 or . k ≡ 3 mod 4, respectively. However, it was noted in [BEW] that it took four more years for Gauss to prove that the sign of the sums is always positive (as stated in Gauss’ diary entry August 30, 1805). In this section, we will highlight a few important results and corollaries on Gauss sums and then apply them to calculate special cases of the Witten-Reshetikhin-Turaev invariants. For an in-depth discussion on Gauss sums, we refer the reader to [BEW]. We will first present the special case, when .m = 1, before introducing the Gauss sum in general terms. 2 See [Ati2] for 2-framings, [BHMV2] for . p1 -structures, and [Walk] for signature and Lagrangian subspaces.

302

16 The Witten-Reshetikhin-Turaev Invariants of 3-Manifolds

Theorem 16.3.1 ([Gau2]) Let k be a positive integer; then k−1 E .

e

2π i k

n=0

n2

⎧ ⎪ ⎪ ⎪ ⎨ ⎪

√

k 0 √ = ⎪ i k ⎪ ⎪ ⎪ (1 + i)√ k ⎩

for k for k for k for k

≡1 ≡2 ≡3 ≡0

mod mod mod mod

4 4 4 4.

(16.4)

An evaluation of the Gauss sum in general terms requires a generalization of the Legendre symbol, known as the Jacobi symbol. ( ) a Definition 16.3.2 The Legendre symbol, denoted by . , where a and p are integers p and p is an odd prime, is defined by ( ) ⎧ ⎪ ⎨ 0 if a ≡ 0 mod p ⎪ a . = 1 if a = 0 mod p and a is a quadratic residue of p ⎪ p ⎪ ⎩ −1 if a = 0 mod p and if a is not a quadratic residue of p where a is a quadratic residue of p if .∃x such that .0 < x < p and .a ≡ x 2 mod p. ( ) a The Jacobi symbol is a generalization of the Legendre with the same notation . n where a is any integer and n is a positive odd integer. Suppose n has the following prime factorization, .n = p1α1 p2α2 · · · pαt t ; then the Jacobi symbol is defined, by a product of Legendre , as

.

( ) αt ( ) ( ) α1 ( ) α2 a a a a ··· . = n p1 p2 pt

Theorem 16.3.3 ([Gau2]) Let m and k be integers such k > 0; then ( ) ⎧ m √ ⎪ ⎪ k for k ≡ 1 ⎪ ⎪ ⎪ k ⎪ ⎪ ⎪ for k ≡ 2 k−1 ⎪ E ⎨ ⎪ ( )0 2π i m 2 m √ . e k n = for k ≡ 3 i k ⎪ ⎪ n=0 ⎪ ⎪ ( )k ⎪ ⎪ k √ ⎪ ⎪ ⎪ ⎪ |m| (1 + i) k for k ≡ 0 ⎩

that .gcd(m, k) = 1 and

.

mod 4 mod 4 mod 4

(16.5)

mod 4.

For the rest of this section, we will assume that A is a 4rth root of unity. The following lemmas serve as useful tools to calculating Gauss sums. The first lemma E 2π i 2 is a standard result usually written as . e c k . k mod c

E E4r (k+j)2 k2 Lemma 16.3.4 Let j be an integer; then . 4r . k=1 A = k=1 A

16.3 Calculations of the Witten-Reshetikhin-Turaev Invariant and Gauss Sums

303

Proof Suppose . j = 4rt for .t ∈ Z; then since . A4r = 1, we have 4r E .

A(k+4rt) = 2

k=1

4r E

Ak

2 +2(4rt)+(4rt)2

=

k=1

4r E

2

Ak .

k=1

Notice that in this case, the fact that the sum was from . k = 1 to 4r was irrelevant. Now suppose . j = 4rt + l where .t, l ∈ Z such that .0 < l < 4r. Then since . A4r = 1 and after an index shift, we have 4r E .

A(k+4rt+l) = 2

k=1

4r E

A(k+l) = 2

k=1

=

4r+l E

2

Ak =

k=1+l

4r E

2

Ak +

k=1+l

l E

4r E k=1+l

2

Ak +

4r+l E

Ak

2

k=4r+1

A(k+4r) . 2

k=1

After applying the first case, noting that in that case it was not required to sum to 4r, we have our desired result. o The next lemma, proved by Robion Kirby and Paul Melvin in [KiMe], is vital to proving the following two theorems. Lemma 16.3.5 ([KiMe]) Let .G = r−2 ( E .

A2l(k+1) − A−2l(k+1)

k=0

=−

)(

4r E k=1

2

Ak .3 Then for any .l, j ∈ Z,

) 2 2 2 A2(k+1)(j+1) − A−2(k+1)(j+1) A(k+1) +(j+1) +l

) G ( 2l(j+1) A − A−2l(j+1) . 2

By using Lemma 16.3.5, Lickorish in [Lic7] proved the following two theorems. Theorem 16.3.6 ([Lic7, Lic8]) Let .G =

4r E

2

An ; then

n=1

Proof We follow the proof given in [Lic8]. We have Recall from Lemma 9.4.6 that a positive kink in a link decorated with the Jones-Wenzl idempotent can be removed by replacing each kink 2 with the product .(−1)n An +2n . Therefore,

.

3 This is a special Gauss sum. We refer the reader to Exercise 16.5.1 for another elegant form.

304

16 The Witten-Reshetikhin-Turaev Invariants of 3-Manifolds

Since A is a 4rth root of unity, then . A2r = −1, and

(16.6)

.

After a bit of work and noticing that . A2r(n+1) = A−2r(n+1) = (−1)n+1 , we can manipulate Eq. 16.6 in the form of Lemma 16.3.5 where .l = r + 1, . k = n, and . j = 0.

.

After applying Lemma 16.3.5, we have

.

By recalling that . A2r = A−2r = −1, we obtain our desired result. Theorem 16.3.7 ([Lic7, Lic8]) Let .G =

4r E

o

A−n ; then 2

n=1

We finish this section with Lisa C. Jeffrey’s result in [Jef] on the WittenReshetikhin-Turaev invariant of lens spaces. Theorem 16.3.8 ([Jef]) The Witten-Reshetikhin-Turaev invariant of the lens space L(p, q), for . p ≥ 1, is given by

.

12s(q, p)(2π i) 1 4r I(L(p, q)) = √ e −2r p ) p ( E qr n 2 (2π i) n(q+1)(2π i) 2π i −2π i qr n 2 (2π i) n(q−1)(2π i) p p − e 2r p e p e e 2r p e p e .

.

n=1

where .s(d, c) =

c−1 E k=1

{ 0, x ∈ Z, and .((x)) = . x − [x] − 1/2, otherwise ( ) ( ) c−1 E πdk . cot πk c cot c

(( ) ) ( ( k c

Alternatively, .s(d, c) =

1 4c

dk c

k=1

))

16.4 The Witten Skein Module

305

16.4 The Witten Skein Module Sikora and Roberts were the first to introduce the notion of taking the quotient of the KBSM of a orientable 3-manifold by the Jones-Wenzl idempotent. In this section, we will give an introduction to this skein module, called the Witten skein module, and discuss an important theorem by Roberts and Sikora: that the Witten skein module depends only on the boundary of the manifold. 2n be Definition 16.4.1 Let M be an orientable 3-manifold with boundary, .{xi }i=1 framed points in .∂ M, .r ≥ 3 be a fixed integer, and A be a primitive 4rth root of unity. Consider the relative Kauffman bracket skein module .S2,∞ (M, {xi }12n ; C(A)) and its submodule generated by all linear skeins consisting of the Jones-Wenzl idempotent 2n . fr−1 , denoted by . Sr−1 (M, {xi } ); then the Witten skein module is defined by 1

SW (M, {xi }12n ; C(A)) := S2,∞ (M, {xi }12n ; C(A))/Sr−1 (M, {xi }12n ).

.

Suppose .L is a framed link in M; then we will denote the element of .L in 2n L >W . .SW (M, {xi } ; C(A)) by . W . Then . in the following way: .

< , > : S2,∞ (D2 × I, {xi }12n ) × S2,∞ (D2 × I, {xi }12n ) −→ R.

Let .ci, c j ∈ CnA . Glue .ci with the inversion of .c j along the marked circle, respecting the labels of the framed points. The resulting picture is that of a disk with disjoint null homotopic circles. Thus, we define . = d m where m denotes the number of these circles. Figure 17.1 illustrates an example of the bilinear form when .n = 4. The Gram matrix of type A is defined as .G nA = ()1≤i, j ≤Cn . Its determinant is called the Gram determinant of type A.

A . Dn

In classical linear algebra, Gram matrices are often defined over inner product spaces. If the determinant of a Gram matrix is not 0, then the vectors on which it is based are

17.2 Background Information

313

linearly independent. If there are k of these vectors in a k-dimensional vector space, then they form a basis for the space.

Fig. 17.1: The bilinear form on two elements in .TL4 . In this case, the result is .d 2 Example 17.2.2 We show a calculation for .G3A (see Table 17.1) and . D3A. We find that A 2 4 4 2 . D = (d − 1) · d · (d − 2d). 3

Table 17.1: The Gram matrix .G3A Recall from Lecture 6 the definition of the Chebyshev polynomial of the second kind. When the free variable is d, we use the notation .Δn . That is: Δ0 (d) = 1, Δ1 (d) = d, Δn (d) = d · Δn−1 (d) − Δn−2 (d). .

Therefore, from Example 17.2.2, we get . D3A = Δ41 · Δ42 · Δ3 .

314

17 Gram Determinants of Type A and Generalized Type A

Proposition 17.2.3 ([Cai]) We have the following: ( 1 a = b+1 θ(a, b, 1) . = Δ a+1 Δa a = b − 1, Δa where .θ(a, b, 1) is the decorated theta curve evaluated in .R2 . See Definition 9.4.9. Proof This follows directly from Lemma 9.4.10 (bubble popping lemma) by making u n the substitution .c = 1.

17.3 The Type A Gram Determinant Formula

Theorem 17.3.1 ([Wes, DiF, Cai]) Let . R = Z[A±1 ]. Then, .

DnA(d) =

n | | Δi αi ( ) , Δi−1 i=1

) ) ( ( 2n 2n A2i+2 − A−2i−2 . − .where Δi = (−1) and αi = n−i−1 n−i A2 − A−2 i

Before proceeding into the technical details, we give a summary of the proof in [Cai]. To compute the Gram determinant . DnA, we change the basis of the TemperleyLieb algebra, .TLn (d), so that in the new basis, the Gram matrix is a diagonal matrix. It follows that the change of basis is given by an upper triangular matrix with 1’s on the diagonal. Therefore, the Gram determinant is unchanged by the change of basis. To find the diagonal entries of the new Gram matrix, we use the standard properties of theta nets and the fact that the trace of the Jones-Wenzl idempotents, in the disk, is the Chebyshev polynomial of the second kind in the variable d. Finally, Cai references [DiF] by using the combinatorial properties of Dyck paths to find a closed formula for the Gram determinant of type A.

17.4 Proof of the Formula Definition 17.4.1 Consider the finite sequence {a1, a2, . . . , a2n−1 } of natural numbers, which satisfies the following two conditions: 1. a1 = a2n−1 = 1.

17.4 Proof of the Formula

315

2. |ai − ai−1 | = 1 ∀i. Let An be the set of all such sequences. The elements of TLn (d), denoted by Da1,a2,...,a2n−1 , are depicted in Fig. 17.2. The triple points in the diagram are admissible (see Lecture 9). This disk could be deformed into a rectangle where the boundary points on the left of the gray line are identified on the left edge of the rectangle and those on the right are identified to the right edge. This is done in such a way we still keep the structure of the Temperley-Lieb algebra.

1

1

1

1

1

1

1

2

Fig. 17.2: The depiction of Da1,a2,...,ak ∈ TLn (d) using a1, . . . , ak where k = 2n − 1 (see Definition 17.4.1)

Example 17.4.2 For n = 3, A3 = {(10101), (10121), (12101), (12121), (12321)}. The diagrams of D3 are pictured in Fig. 17.3:

1,2,3,2,1

1,0,1,0,1

1,2,1,2,1

1,0,1,2,1

1,2,1,0,1

Fig. 17.3: The set D3

There is a formula for the inner product of two elements in Dn . Theorem 17.4.3 ([Cai]) .

= δa1,b1 . . . δa2n−1,b2n−1

θ(a2n−1, a2n−2, 1) θ(a2, a1, 1) ... Δa2n−1 . Δ a2 Δa2n−1

316

17 Gram Determinants of Type A and Generalized Type A

1 1

...

... Fig. 17.4: A depiction of , where k = 2n − 1

Proof Consider the schematic diagram in Fig. 17.4. Suppose there is an index i such that bi = ai . Then there is a return into the Jones-Wenzl idempotent at that index. In this case, the diagram evaluates to 0. This is why the δ factors are included in the formula. Suppose the two sequences are equal. We proceed by induction on n. A direct calculation shows the initial condition for n = 2. Suppose the formula holds for all N < n. To show it holds for the case n, pop the first bubble using Lemma 9.4.10. This yields the following: .

=

θ(a2, a1, 1) . Δ a2

Using our inductive hypothesis, we obtain our result.

u n

Notice that Theorem 17.4.3 implies that our Gram matrix with respect to the collection Dn will be diagonal. Thus, multiplying the diagonal elements computes the Gram determinant with respect to this collection. The closed formula above computes the multiplication of diagonal entries. Using properties of Chebyshev polynomials and the Jones-Wenzl idempotent, every element of CnA can be written as a linear combination of elements in Dn . For details, see [Cai]. The following theorem summarizes this result. Theorem 17.4.4 ([Cai]) Dn is a basis for TLn (d). The change of basis matrix between CnA (the set of all crossingless diagrams) and Dn is upper triangular with 1' s on the diagonal. Therefore, this matrix has determinant 1. So we have | | | | A| A| . Dn | = Dn | CnA

Dn

=

| | θ(a2, a1, 1) θ(a3, a2, 1) θ(1, a2n−2, 1) ... Δ1 . Δ Δ Δ1 a2 a3 a ∈A n

n

(17.1)

17.4 Proof of the Formula

317

The formula above simplifies to that in Theorem 17.3.1. We show this using a combinatorial argument involving Dyck paths (following [Cai] and [DiF]). Consider the following illustrative example. Notice when the sequence steps down from n to n − 1 at some index, the factor appears. We must count how many times across all admissible sequences such a step-down occurs. The following set accomplishes this enumeration. Δn Δ n−1

Definition 17.4.5 Define the set Gk to be the set of all tuples (Da∗ , i) that consists of a basis element with an admissible sequence that has a step from k to k − 1 at index i. Let ak = |Gk |; then

n | | Δk a k ( ) . Δk−1 k=1 ( 2n ) ( 2n ) − n−k−1 . Therefore, it suffices to prove that ak = n−k .

DnA =

Definition 17.4.6 A lattice path is a path from the origin to a specified point (a, b) ∈ Z × Z. The walk diagram is allowed to go up or down by one unit for each horizontal unit. A Dyck path is a lattice path that never goes below the x-axis. Let D(a,b) denote the set of all Dyck paths from the origin to the point (a, b). We define a bijection between Gk and the subset of D(2n,2m) , where there is a step from k to k − 1 at some index i. We send the element D(a1,a2,...,a2n−1 ) to the path that has a height ai at step i. At step 2n, the path returns to 0. See Fig. 17.5.

10121

→

Fig. 17.5: The Dyck path corresponding to the basis element D10121

This is a bijection since the path is completely determined by the sequence and each path gives rise to a unique sequence. So counting |Gk | is the same as counting the number of such Dyck paths. Further, the number of Dyck paths from (0, 0) to (2n, 0) with a step from k to k − 1 is in bijection with the set D(2n,2m) . Consider the illustration of the correspondence given in Fig. 17.6. Given a Dyck path D ∈ D(2n,2m) , at some point, the path crosses the line y = m in an upward slope. Cut the graph of the path at the rightmost of these points. Flip

318

17 Gram Determinants of Type A and Generalized Type A

cut here

reflect

shift

Fig. 17.6: An illustration of converting a path in D(2n,2m) to a path in D2n,0

the path about its vertical axis, and then shift down by m units so that the rightmost point is identified with the rightmost endpoint of the cut. Glue the path back together. Now you have a path in D(2n,0) . This path will have a step-down from m to m − 1 at some point to the right of the cut. Let the rightmost of these points be the index of the moves. By the reflection the tuple in Gm . This process is invertible ( 2n ) by( reversing 2n ) − n−m−1 . property, we have that |D(2n,2m) | = n−m This concludes the proof (following Cai) that DnA(d) = ( 2n ) ( 2n ) 2i+2 −2i−2 − n−i−1 . (−1)i A A2 −A and αi = n−i −A−2

n | | i=1

i ( ΔΔi−1 )αi , where Δi =

17.5 The Generalized Type A Gram Determinant We now turn our attention to a different bilinear form introduced in [BIMP1]. The basis elements are once again the crossingless connections on 2n boundary points on a disk. We construct the new bilinear form as follows. Definition 17.5.1 Let the disk . D2 , with 2n marked points on its boundary, be considered as a rectangle with n points on the top edge and n points on the bottom edge. Define a bilinear form . < , > in the following way: .

< , > : S2,∞ (D2 × I, {xi }12n ) × S2,∞ (D2 × I, {xi }12n ) −→ Z[d, z].

As commented in Lecture 8, in this section, we will be viewing the elements of TLn vertically and multiplying from top to bottom. For .ai, a j ∈ A n , glue .ai with the reflection about the horizontal axis of .a j , which is denoted by .a j , such that the bottom edge of .ai is identified with the top edge of .a j . Connect the marked points on the top edge of .ai with those on the bottom edge of .a j , in the annulus, respecting the ordering of the marked points (see Fig. 17.7). Recall that the described operation is the same as taking the trace of the image of the bilinear form.

.

The result is an annulus with two types of disjoint circles, homotopically trivial and nontrivial. Thus, we define . = d k z m where k and m denote the number of these circles, respectively. We define the Gram matrix of generalized type A as Agen = () Agen .G n i j 1≤i, j ≤Cn and denote its determinant by . Dn .

17.5 The Generalized Type A Gram Determinant

319

Fig. 17.7: The bilinear form of generalized type A

Example 17.5.2 Consider the basis of Catalan connections on six boundary points. gen Figure 17.8 illustrates the calculation of . . The matrix .G3A is computed in Table 17.2, along with its determinant.

Fig. 17.8: The bilinear form of generalized type A Let .Δn (d) and .Sn (z) denote the nth Chebyshev polynomials of the first kind in the variables d and z, respectively. We calculate and factor the determinant: gen

.

D3A

= (d 2 − 1)4 z 5 (z2 − 2)

(17.2)

= Δ2 (d) S3 (z)S1 (z) .

(17.3)

4

4

The Gram determinant of generalized type A is a polynomial that can always be written as a product of ratios of Chebyshev polynomials. Theorem 17.5.3 ([BIMP1])

gen

.

DnA (d, z) = DnA(d)

( )2 n ) (ni )−(i−1 [ n2 ] ( ) | | Sn−2i (z) i=0

Δn−2i (d)

.

The proof of this result in [BIMP1] closely follows the proof in [Cai] for type A. We change the basis of .TLn (d) so that under the new basis, the Gram matrix is diagonal. In fact, for generalized type A , this is the same basis as for type A (see Definition 17.4.1). Therefore, the change of basis is upper triangular with .1' s along the diagonal, and the Gram determinant does not change as a result of the change of basis. This reduces the problem to calculating the diagonal entries of the matrix under the new basis.

320

17 Gram Determinants of Type A and Generalized Type A

gen

Table 17.2: The Gram matrix .G3A

The proof of Theorem 17.4.4 also holds when the generalized type A bilinear form is used. Corollary 17.5.4 The Gram determinant of generalized type A is unchanged under the change of basis from .CnA to .Dn . This matrix therefore has determinant 1. The following formula efficiently computes the diagonal entries of the generalized type A Gram matrix. Theorem 17.5.5 ([BIMP1]) .

G en = A

San (z) . Δan (d)

Proof Figure 17.9 visualizes . Gen . We proceed by bubble popping (using Lemma 9.4.11 and Corollary 9.4.12). So we have the following:

17.5 The Generalized Type A Gram Determinant

321

Fig. 17.9: Bubble reduction. On line 1, the hatted factor indicates that the terms with the index under the hatted factor have been omitted

gen

.

DnA

=

| |

G en

{a n } ∈ A n

=

| |

A

{a n } ∈ A n

= DnA

| |

{a n } ∈ A n

Sa n Δan

Sa n . Δan

We now show that the number of admissible with a middle term .an = n−2i, ) 2 ( ) ( n sequences ) . Due to the Désiré André reflection where .0 ≤ i ≤ n/2, is equal to .( ni − i−1 principle the number of sequences of length n that end in .an = n − 2i is equal ( ) ( [AD], n ) . Pairing these with all sequences of length n that start with .a1 = n − 2i to . ni − i−1 ( ) (n) 2 S (z) ) . u n gives the desired result. Hence, the exponent on . Δa na is .( ni − i−1 n

Lemma 17.5.6 ([BIMP1]) There are .

(n) i

−

(

n i−1

)

Dyck paths from .(0, 0) to .(n, n − 2i).

( ) Proof There are . ni such paths without the restriction. To travel from 0 to .n − 2i in n steps, the graph must take a step-down i times. We proceed by subtracting the bad paths that dip below the x-axis. Suppose you have such a path. Travel along the path from .(0, 0) until you reach the point with a y-coordinate of .−1 for the first time. Reflect the path prior to this point ( nabout ) . y = 1. See Fig. 17.10. The new path connects ( ) .(0, ( n −2) ) to .(n, n − 2i). There such paths so the number of valid paths is . ni − i−1 . are . i−1 The lemma further simplifies the calculation of the Gram determinant under the generalized type A bilinear form. This completes the proof of the formula:

322

17 Gram Determinants of Type A and Generalized Type A

−1

Fig. 17.10: Dyck paths and their reflection about the horizontal axis . y = −1

gen

.

DnA (d, z) = DnA(d)

( )2 n ) (ni )−(i−1 [ n2 ] ( ) | | Sn−2i (z) i=0

Δn−2i (d)

. u n

17.6 Conclusion Gram determinants continue to be an open area of research in knot theory. By defining new bilinear forms or changing the ambient manifold of the Kauffman bracket skein module, novel algebraic structures can be analyzed. An interesting example is given in [FrKa], where the bilinear form is defined over the topological quantum field theory of surfaces into the Witten-Reshetikhin-Turaev invariant of 3-manifolds glued along boundary surfaces. We describe constructions of the Gram determinant based on the annulus and the Möbius band in the next lecture.

17.7 Exercises Exercise 17.7.1 Draw the basis of Catalan connections on eight boundary points. Express this basis pictorially in the standard way and in terms of admissible sequences. See Example 17.4.2. Exercise 17.7.2 Compute the type A and generalized type A inner product for D1,2,3,2,1 and D1,0,1,2,1 in two ways – pictorially and by applying Theorems 17.4.3 and 17.5.5. Exercise 17.7.3 Show the base case calculation for n = 2 in the proof of Theorem 17.4.3. Exercise 17.7.4 Use Formula 17.1 to calculate the Gram determinant for n = 3. Exercise 17.7.5 List the elements of the set G2 in D3 (see Definition 17.4.5).

Lecture 18 Gram Determinants of Type B and Type Mb

In this lecture, the Gram determinant of the type B Temperley-Lieb algebra is introduced. We present a proof of the closed formula. The proof was originally given by Qi Chen and the first author and influenced by the work of W. B. Raymond Lickorish on the Witten-Reshetikhin-Turaev invariant of 3-manifolds. We then discuss the Gram determinant of type Mb (Möbius band). A closed formula for this Gram determinant was originally conjectured by Chen in 2009. Almost 10 years later, the first three authors, along with Sujoy Mukherjee, proved several facts supporting Chen’s hypothesis. At the end of the lecture, we discuss these results.

18.1 Introduction As mentioned in the previous lecture, it is possible to construct different bilinear forms by changing the ambient surface of the Kauffman bracket skein module . This lecture investigates the Gram determinant using the annulus and the Möbius band. Rodica Simion investigated bilinear forms of type B in her work on chromatic joins (see [Sim, Sch, ChP1]). Mieczysław K. Dabkowski and the first author hypothesized that the Gram determinant is related to Chebyshev polynomials of the first kind, and Gefry Barad conjectured the complete closed formula for the determinant. This formula was proven by Qi Chen and the first author in [ChP2]. The proof in [ChP2] uses basic combinatorics and properties of Jones-Wenzl idempotents. Paul Martin and Hubert Saleur were the first to consider the type B Gram determinant. Their work focuses on applications to statistical mechanics, and they employ representation theory to prove their determinant in [MaSa1, MaSa2]. The proof by Martin and Saleur and that of Chen and the first author employ different strategies and are both of interest. In this lecture, we follow [ChP2].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_18

323

324

18 Gram Determinants of Type B and Type Mb

The first author proposed the notion of the Gram determinant of type Mb. In April 2009, Chen gave a hypothesis for the closed formula computing the Gram determinant of the Möbius band. This vein of research was pursued almost 10 years later in [BIMP1]. Although the conjecture is still unproven, recent partial results supporting it are outlined in the second part of this lecture.

18.2 The Gram Determinant of Type B Consider an annulus with 2n marked points along the outer boundary. See Fig. 18.1. 2 −1

1

2

Fig. 18.1: An annulus with 2n boundary points As before with our type A determinants, we can connect these 2n points to form crossingless connections in the Let . Bn denote the set of these diagrams. In ( annulus. ) such diagrams. See Fig. 18.2, which illustrates . B2 . Lecture 7, we show there are . 2n n

1

2

3

4

5

6

Fig. 18.2: The set . B2 We define a bilinear form on these annular diagrams as follows: Definition 18.2.1 Let . A2 be an annulus with 2n marked points on its outer boundary. Let . Bn = {b1, b2, . . . , b(2n) } be the set of all diagrams of crossingless connection n between these 2n points. Define the type B bilinear form . < , > in the following way: .

< , > : S2,∞ (A2 × I, {xi }12n ; R, A) × S2,∞ (A2 × I, {xi }12n ; R, A) −→ R[d, z].

18.2 The Gram Determinant of Type B

325

Given .bi, b j ∈ Bn , glue .bi with the inversion of .b j along the marked circle, respecting the labels of the marked points. The resulting picture has disjoint circles, which are either homotopically nontrivial or null homotopic. Then, . = z k d m , where k and m denote the number of these circles, respectively. The Gram matrix of type B is defined as .G nB = ()1≤i, j ≤(2n) . Its determinant

.

DnB is called the Gram determinant of type B.

n

Example 18.2.2 The bilinear form . is illustrated in Fig. 18.3.

Fig. 18.3: The inner product of two elements in . B4 . In this case, the result is dz

Example 18.2.3 Table 18.1 illustrates the Gram matrix .G2B . We calculate the Gram determinant for the collection . B2 . We calculate the determinant of this matrix to be .

D2B (d, z) = −(d − z)4 (d + z)4 (−2 + d 2 + z)(−2 + d 2 + z) = (d 2 − z 2 )4 ((d 2 − 2)2 − z 2 ).

Recall from Lecture 6 that .Tn denotes the Chebyshev polynomial of the first kind defined recursively by the equation .Tn+1 (d) = d · Tn (d) − Tn−1 (d), with the initial conditions .T0 (d) = 2 and .T1 (d) = d. In Exercise 18.4.2, we see that the determinant given above can be expressed in terms of Chebyshev polynomials. In general, it is possible to rewrite the Gram

326

18 Gram Determinants of Type B and Type Mb

Table 18.1: The Gram matrix .G2B

determinant of type B in terms of Chebyshev polynomials of the first kind for all n. The following closed formula was proven in [ChP2] and answers Simion’s question. Theorem 18.2.4 ([ChP2, MaSa2]) Let . R = Z[A±1 ]; then . DnB =

n | | i=1

(Ti (d)2 − z 2 )(n−i ) . 2n

Proof We sketch the proof ( )for the Gram determinant of type B given in [ChP2]. For a given n, we have . 2n n Catalan states in the annulus. The evaluation of the Gram matrix of type B at . z = (−1)k−1Tk (d) is closely related to the Gram matrix when the hole in the annulus is decorated by the kth Jones-Wenzl idempotent . fk . More precisely, we form a Hopf link with one component the annulus and a new component decorated by the kth Jones-Wenzl idempotent. After this decoration, the ( ) ( 2n ) number of Catalan states is equal to . 2n n − n−k . Thus, the nullity of the matrix is ( ) 2n at least . 2n . As a result, . D B is divisible by .(Tk (d) + (−1)k z)(n−k ) . Therefore, the n−k

n

theorem holds up to a multiplicative factor. By analyzing the highest-degree term, coming from the diagonal, we see that this factor is equal to 1. See [ChP2] for more o details.

18.3 The Gram Determinant Based on the Möbius Band

327

18.3 The Gram Determinant Based on the Möbius Band The Klein bottle is the connected sum of two projective planes. Equivalently, it is obtained by gluing two Möbius bands along their boundaries. It is illustrated in Fig. 18.4, where the red line denoted by z separates the Möbius bands. These Möbius bands are usually called crosscaps, and in Fig. 18.4, there are two: one lies in the center and one is along the outside “edge” of the figure. There are five homotopically distinct curves on the Klein bottle, as seen in Fig. 18.4. The trivial curve is denoted by d and the separating curve by z. The curve that intersects both crosscaps is w. The core of the inner Möbius band is x, and that of the outer is y. See [Lic8] and Appendix B for more information on the homotopically distinct curves in the Klein bottle.

Fig. 18.4: The five different simple closed curves on the Klein bottle This structure allows us to define a bilinear form on the Möbius band. Consider a Möbius band with 2n marked points on its boundary and n crossingless curves connecting these 2n marked points. Denote the set of all diagrams of crossingless connections on the Möbius band with 2n points by .Mbn . Then the number of distinct crossingless connections is .

|Mbn | =

n ( ) E 2n k=0

Notice that in the formula, .

n ( ) E 2n k=0

k

k

=2

2n−1

( ) 1 2n . + 2 n

, the index k tells us how many curves are disjoint

from the crosscap (see Fig. 18.6). See Exercise 7.2.6 and Corollary 13.5.2.

328

18 Gram Determinants of Type B and Type Mb

Definition 18.3.1 Define a bilinear form . < , > on the crossingless connections in the Möbius band in the following way: .

< , > : S2,∞ (Mb ׈ I, {xi }12n ) × S2,∞ (Mb ׈ I, {xi }12n ) −→ Z[d, w, x, y, z],

where .d, w, x, y, and z denote the five homotopically distinct simple closed curves on the Klein bottle, which are illustrated in Fig. 18.4. Given .mi, m j ∈ Mbn , identify the boundary component of .mi with that of the inversion of .m j , respecting the labels of the marked points. The result is a collection of disjoint simple closed curves on the Klein bottle, which are of five different types as shown in Fig. 18.4. In particular, given two elements in .Mbn , the image of its bilinear form is a monomial in .Z[d, x, y, z, w], that is, . = d m x n y k zl w h where .m, n, k, l, and h denote the number of these simple closed curves, respectively. See Fig. 18.5 for an example of the calculation. The Gram matrix of type Mb is Mb defined as . GMb n = ()1≤i, j ≤ |Mb n | , and its determinant is denoted by . Dn . See [IM] for a different Gram determinant from the Möbius band, which has connections to a new type B Gram determinant.

Fig. 18.5: The bilinear form of type .Mb for two elements of . B4

Exercise 18.3.2 Show that .

n ( ) E 2n k=0

k

= 22n−1 +

( ) 1 2n 2 n .

Given a collection of crossingless connections in the Möbius band, we can calculate the Gram matrix and determinant, as in Example 18.6. Example 18.3.3 In [BIMP1], . D2Mb is calculated. There are 11 crossingless connections in the Möbius band when 2 boundary points are used. These are illustrated in Fig. 18.6. The Gram matrix given by the bilinear form is illustrated in Table 18.2. The determinant of the Gram matrix is

18.3 The Gram Determinant Based on the Möbius Band

329

Fig. on the Möbius band, the first ( )18.6: In the 11 different crossingless connections ( ) 6, . 42 , do not intersect the crosscap; the next 4, . 41 , pass through the crosscap once; ( ) and, finally, the last 1, . 40 , passes through the crosscap twice

Table 18.2: The Gram matrix .GMb 2

.

D2Mb = (d − z)4 [(d + z)w − 2xy]4 (d 2 (d 2 − 4))(d 2 − 2 + z) ×[(d 2 − 2 − z)(w 2 − 2) − 2(2 − z)] = (T1 (d) − z)4 [(T1 (d) + z)T1 (w) − 2xy]4 (T4 (d) − 2)(T2 (d) + z) ×[(T2 (d) − z)T2 (w) − 2(2 − z)].

These calculations support the following conjecture by Chen.

330

18 Gram Determinants of Type B and Type Mb

Conjecture 18.3.4 (Chen) Let . R = Z[A±1, w, x, y, z]. Then the Gram determinant of type Mb for .n ≥ 1, denoted by . DnMb , is .

DnMb (d, w, x, y, z) =

n | |

(Tk (d) + (−1)k z)(n−k ) 2n

k=1 n | |

((Tk (d) − (−1)k z)Tk (w) − 2x y)(n−k )

k=1 k odd n | |

2n

((Tk (d) − (−1)k z)Tk (w) − 2(2 − z))(n−k ) 2n

k=1 k even n | |

Dn,i,

i=1

where . Dn,i =

n | | k=1+i

(T2k (d) − 2)(n−k ) and i represents the number of curves passing 2n

through the crosscap. Using a computer program, the conjecture was verified for .n = 2, 3, and 4. In the remaining part of this lecture, we discuss some recent work on the Gram determinant of type Mb and prove a few results supporting Conjecture 18.3.4 (Propositions 18.3.5, 18.3.6, 18.3.7, Theorems 18.3.8, and 18.3.9). Proposition 18.3.5 ([BIMP2]) . DnMb is divisible by .(d − z)(n−1) . 2n

We leave the proof as an exercise for the reader. Proposition 18.3.6 ([BIMP1]) . DnMb is divisible by .(w(d + z) − 2xy)(n−1) . 2n

Proof Consider the set .A consisting of the three crossingless diagrams given in Fig. 18.7. The diagrams .b1 and .b2 are constructed from .b3 by pushing the connection off the central crosscap. The strings above and below this connection combine in the same way for all of the bilinear forms in the Gram matrix generated by these three elements. These form the monomial u in front of each matrix entry given below. ⎡ ud uz uy ⎤ ⎢ ⎥ .G = ⎢ uz ud uy ⎥ . ⎢ ⎥ ⎢ ux ux uw ⎥ ⎣ ⎦ Replace the third column of this matrix (.c3 ) with .(d + z)c3 − y(c1 + c2 ) to obtain the following: ⎡ ud uz ⎤ 0 ⎢ ⎥ ∗ ⎥. 0 .G = ⎢ uz ud ⎢ ⎥ ⎢ ux ux u(w(d + z) − 2xy) ⎥ ⎣ ⎦

18.3 The Gram Determinant Based on the Möbius Band

331

Further, .det(G∗ ) = (d + z) det(G). When you multiply any element c of .Mbn by .b1 , you obtain ud, ux, or uz (here, u is a monomial) depending on the connection for that node in c. Similarly, multiplying by .b2 yields z, d, or x and by .b3 yields y or w. As such, the column block under .b1 , .b2 , and .b3 consists of entries proportional (by a monomial) to .(d, z, y), .(z, d, y), or .(x, x, w). Thus, the column operation that replaces the column under .b3 with .(d + z)b3 − y(b1 + b2 ) gives a( 0 or) a multiple of 2n elements that .−2xy + d(w + z). This operation can be performed for each of the . ( 2n ) n−1 touch the crosscap exactly once. So we have a matrix with . n−1 columns divisible by .((d + z)w − 2xy). Further, .(d + z) and .((d + z)w − 2xy) are coprime. Thus, we obtain the desired result. o

Fig. 18.7: Three elements in . M bn that differ by one curve

Proposition 18.3.7 . DnMb is divisible by .((d 2 − 2 − z)(w 2 − 2) − 2(2 − z))(n−2) . 2n

( 2n ) Proof Consider 2n points on the boundary of a Möbius band. There are . n−2 crossingless connections with exactly two arcs going through the crosscap. Let b be one of them with arcs going through the crosscap denoted by .α1 and .α2 . Each such b leads to six states in the annulus obtained by deforming .α1 and .α2 to be disjoint from the crosscap. See Fig. 18.8.

Fig. 18.8: The class defined by b

These seven elements produce the .7 × 7 block in the Gram matrix illustrated in Table 18.3. Consider the column:

332

18 Gram Determinants of Type B and Type Mb

Table 18.3: A .7 × 7 block of the Gram matrix

.

b' = b1 + 2b3 + b4 + b6 − b2 − b5 + (d 2 − 2 − z)b = (0, 0, 0, 0, 0, 0, α(d 2 − 2 − z)(w 2 − 2)(2(2 − z)))T .

Observe that for any state .m ∈ Mb2 , . , , . . . , , and . is proportional to this .7 × 7 block. In effect, the column under .b' is divisible by .(d 2 − 2 − z)(w 2 − 2) − 2(2 − z). We can perform the( same operation we did to b to any element that ( 2ncuts ) the crosscap twice. 2n ) of these. Thus, we obtain a matrix with . n−2 columns divisible by There are . n−2 2 2 2 2 .(d − 2 − z)(w − 2) − 2(2 − z). Noting additionally that . d − 2 − z and .(d − 2 − 2 o z)(w − 2) − 2(2 − z) are coprime, we obtain our result. The pattern to generalize these three propositions is somewhat clear; ( 2nhowever, ) states implementing the strategy is difficult. One would start with any of the . n−k with k arcs that go through the crosscap. This crosscap is resolved in all possible ways. The new states have .n − k fewer crosscap cutting arcs. The resulting states give a relation with monomial coefficients. Finding this relation is where the difficulty of the generalization lies. Theorem 18.3.8 ([BIMP1]) .

n | |

(Tk (d) + (−1)k z)(n−k ) divides . DnMb .

k=1

2n

18.3 The Gram Determinant Based on the Möbius Band

333

Hint: Adapt the proof for type B given in [ChP2]. Proceed ( ) analogously, by i−1T (d), w) is at least . 2n . (d, x, y, (−1) showing the nullity of .GMb i n n−i ( )2 ( ) Mb Theorem 18.3.9 ([BIMP1]) . Dn−1 divides . DnMb , if . d 2 − 1 is invertible. Proof The proof closely follows that of the main result from [PZ]. There are a few key differences. There are several embeddings of .Mbn−1 into .Mbn . The argument relies on the map .ik,k+1 : Mbn−1 → Mbn defined by inserting a cup from k to . k + 1. See the diagram below (Fig. 18.9). 1

2 −2 2 −3 1,2

: 1 2

...

→

2 2 −1

2

...

Fig. 18.9: The embedding map .i1,2 Mb ). Let . b ∈ BMb ; then there In the Gram matrix, consider the block under .i2n,1 (Bn−1 n ' Mb is an element .b ∈ i1,2 (Bn−1 ) such that the block row . is proportional to . , is a function from the set of unoriented link diagrams .D to Laurent polynomials with integer coefficients in the variable A, . < > : D −→ Z[A±1 ]. The polynomial is characterized by the initial conditions . = 1 and . = (−A2 − A−2 ) and the skein relation .

Recall that the bracket polynomial is not a link invariant as it is affected by Reidemeister moves of the first type. However, it is an invariant of framed links. For a detailed discussion of the matter, refer to Lecture 5. Below, we define the notion of a Kauffman state, and we use it to write the polynomial as a Kauffman state sum.

19.2 The Kauffman Bracket Polynomial

337

Definition 19.2.2 Let D be an unoriented link diagram and .cr(D) the set of its crossings. A Kauffman state s, of D, is a function .s : cr(D) −→ {A, B}. We interpret this function as an assignment of a marker to each crossing according to the convention illustrated in Fig. 19.1. Denote by KS the set of all Kauffman states. Moreover, every marker yields a natural smoothing of the crossing.

Fig. 19.1: Markers at a crossing v of D

Thus, the Kauffman bracket polynomial of the link diagram D is given by the following state sum formula: .

=

E

A |s

−1 (A) |− |s −1 (B) |

(−A2 − A−2 ) |Ds |−1,

s e KS

where . Ds denotes the system of circles obtained after the smoothing of all crossings of D according to the markers of s and . |Ds | denotes the number of circles in the system. In order to define the Khovanov homology, we use the unreduced version of the KBP defined as follows. Definition 19.2.3 The unreduced Kauffman bracket polynomial is a version of the KBP normalized to be 1 for the empty link .∅. It is denoted by .[ ], and with this notation, we have .[∅] = 1, .[O] = (−A2 − A−2 ), and .[D] = (−A2 − A−2 ). The previous definition implies that the Kauffman bracket state sum formula becomes E −1 −1 .[D] = A |s (A) |− |s (B) | (−A2 − A−2 ) |Ds | . s e KS

Notice that the terms of the sum are coming from the states of the diagram and thus they have a geometric interpretation. Moreover, since our goal is to construct a chain complex whose homology is a link invariant, Kauffman states cannot be considered as counterparts of simplices in the construction of simplicial homology: a state contributes a polynomial which is not a monomial. Furthermore, by having the state sum formula from the unreduced version of the Kauffman bracket polynomial, it is possible to define a one-to-one correspondence between the circles in . Ds and the factors .(−A2 − A−2 ). This leads to the idea of an enhanced Kauffman state, as defined by Viro.

338

19 Khovanov Homology: A Categorification of the Jones Polynomial

19.3 Enhanced Kauffman States and Basis for KH Definition 19.3.1 An enhanced Kauffman state S of D is a Kauffman state s together with a function ε : Ds −→ {+, −}, assigning to each circle of Ds a positive or a negative sign. Denote by EKS the set of all enhanced Kauffman states. For a Kauffman state s, there are 2 |Ds | enhanced Kauffman states. We can now express the Kauffman bracket polynomial as a sum of monomial terms coming from the enhanced Kauffman states as follows: E −1 −1 −1 −1 .[D] = (−1) |Ds | A |s (A) |− |s (B) | (A2 ) |ε (+) |− |ε (−) | . S e EKS

Let σ(s) be the difference between the number of positive and negative markers in the state, and let τ(S) be the difference between the number of positive and negative signs in the enhanced state. That is, σ(s) = |s−1 (A)| − |s−1 (B)|, and τ(S) = |ε −1 (+)| − |ε −1 (−)|. Then, we obtain E

[D] =

.

(−1) |Ds | Aσ(s)+2τ(S),

S e EKS

which is the enhanced Kauffman state sum formula for the unreduced KBP. The enhanced Kauffman states form a basis for the chain groups of the Khovanov chain complex. We now define the bigrading on EKS, the chain groups, and the differentials (i.e., boundary maps). Definition 19.3.2 The bidegree on the enhanced Kauffman states is defined as Sa,b (D) = Sa,b = {S e EKS | a = σ(s), b = σ(s) + 2τ(S)} .

.

The chain groups Ca,b (D) = Ca,b are defined to be the free abelian O groups with basis Sa,b (D) = Sa,b , i.e., Ca,b = ZSa,b . Therefore, C(D) = Ca,b (D) is a a,b e Z

bigraded free abelian group. Definition 19.3.3 For a link diagram D, we define the chain complex C(D) = {( )} Ca,b, ∂a,b , where the differential map ∂a,b : Ca,b −→ Ca−2,b is defined by .

∂a,b (S) =

E

'

(−1)t(S,S ) (S, S ')S ' .

S' e Sa−2, b

Here, S e Sa,b , and (S, S ') is the so-called incidence number of S and S '. Specifically, either (S, S ') = 0 or (S, S ') = 1, and it is equal to 1 if and only if the following conditions hold:

19.3 Enhanced Kauffman States and Basis for KH

339

1. S and S ' are identical except at only one crossing, say v. Moreover, s(v) = A and s '(v) = B, where S and S ' are enhancements of s and s ', respectively. 2. τ(S ') = τ(S) + 1, and every component of Ds , not interacting with the crossing v, keeps its sign for Ds' . In particular, condition (1) reflects the fact that the value of σ is decreasing by 2, while condition (2) indicates that either the number of negative signs decreases ' or the number of positive signs increases. Finally, (−1)t(S,S ) requires an ordering of the crossings in the link diagram D. Specifically, t(S, S ') is defined as the number of crossings of D with B markers in S bigger than v in the chosen ordering. This condition is sufficient for the differential to satisfy ∂a−2,b ◦ ∂a,b = 0. It is important to remark that the homology does not depend on the ordering of crossings. Following the notation and definitions so far introduced, we can geometrically list all possibilities for the enhanced states S and S ' to be incident, i.e., (S, S ') = 1. Figure 19.2 describes these cases. Notice that |DS' | = |DS | ± 1. For instance, in the case where two circles each with a negative sign are joined together, the resulting circle must have a negative sign.

Fig. 19.2: Signs of circles after fusion and split of components

Definition 19.3.4 The Khovanov homology of the diagram D is defined to be the homology of the chain complex C(D): .

Ha,b (D) =

ker(∂a,b ) . im(∂a+2,b )

340

19 Khovanov Homology: A Categorification of the Jones Polynomial

From the definition, it follows that the maximum value of a, amax , is |cr(D)| and amin = −|cr(D)|. Similarly, bmax = |cr(D)| + 2|Ds A | and bmin = −|cr(D)| − 2|DsB |, where s A (respectively, sB ) denotes the state s having an A (respectively, B) marker at all crossings. For every diagram D, we see that Ca,b (D) = 0 for a > |cr(D)|, a < −|cr(D)|, b > |cr(D)| + 2|Ds A |, or b < −|cr(D)| − 2|DsB |. It is always the case that Camax,bmax = Z = Camin,bmin , but it often happens that H∗,bmax = 0 or H∗,bmin = 0 (see Lecture 20). Remark 19.3.5 One of the interesting facts of Khovanov homology is its relation with topological quantum field theories (TQFT) and Frobenius algebras. Indeed, it is possible to consider the signs of the circles as generators of a commutative Frobenius algebra A. The fusion of two components into one component can be thought as a multiplication m : A ⊗ A −→ A. Similarly, the splitting of one circle can be understood as comultiplication Δ : A −→ A ⊗ A. In our settings, from Fig. 19.2, we have that the multiplication m is given by the following Table 19.1: m −+ − −+ + + 0 Table 19.1: Table of the multiplication m On the other hand, the comultiplication Δ is given by Δ(+) = (+, +) and Δ(−) = (+, −) + (−, +). Finally, to make the table “more convincing”, we rotate the signs + and − to obtain x and 1, respectively. Then, the multiplication m is given by Table 19.2. The comultiplication now has the form Δ(1) = (x, 1) + (1, x), Δ(x) = (x, x), which describes the Frobenius algebra Zx[x] 2 used by Khovanov. m

1

x

1 x

1 x

x 0

Table 19.2: Table of the multiplication m To conclude this remark, Fig. 19.3 illustrates the conditions that define a Frobenius system F (not including unit and counit). Here, (F, +) is a k-module, and F is equipped with a (commutative) multiplication operation μ : F ⊗ F → F and a comultiplication operation Δ : F → F ⊗ F. Both operations satisfy the associativity and coassociativity properties, namely, μ(μ⊗ Id) = μ(Id ⊗ μ) and (Δ⊗ Id)Δ = (Id ⊗Δ)Δ. Moreover, the Frobenius condition is also satisfied, i.e., Δμ = (Id ⊗ μ)(Δ ⊗ Id).

19.4 Translation to Classical Khovanov (Co)homology

341

Fig. 19.3: Frobenius system conditions

For a more detailed exposition of these ideas, the reader is referred to [Kho1, BarN1]. Surely, the most important property of Khovanov homology as we defined is its invariance under the second and third Reidemeister moves. The following theorem summarizes the results. Theorem 19.3.6 Let D be a link diagram. With the notation given above, the homology groups ker(∂a,b ) . Ha,b (D) = im(∂a+2,b )

are invariant under Reidemeister moves of second and third types. Therefore, they are invariants of unoriented framed links. Moreover, the effect of the first Reidemeister move (positive or negative) R1 is the shift in the homology, Ha,b (R1+ (D)) = Ha+1,b+3 (D) and Ha,b (R1− (D)) = Ha−1,b−3 (D). These groups categorify the unreduced Kauffman bracket polynomial and are called the framed Khovanov homology groups [Kho1, Vir1].

19.4 Translation to Classical Khovanov (Co)homology Khovanov originally associated a bigraded chain complex with an oriented link diagram whose homology is a link invariant. As it was previously mentioned, Kauffman in 1985 introduced the bracket polynomial for unoriented links with the remarkable result that the Jones polynomial can be easily obtained from it by additionally considering an orientation on the diagram; see Lecture 5.

342

19 Khovanov Homology: A Categorification of the Jones Polynomial

- be an oriented link diagram obtained after choosing an orientation In effect, let . D - = w be its writhe. Then, the classical on the unoriented link diagram D. Let .w( D) i, j Khovanov (co)homology, .H ( D), and the framed version of KH, .Ha,b (D), are related by the following equalities: 3w−b - = Hw−2i,3w−2j (D) = Ha,b (D) = H w−a 2 , 2 H i, j ( D) ( D).

.

The following example illustrates the process of obtaining the Khovanov homology of a link. Example 19.4.1 We partially compute the Khovanov homology of the right-handed trefoil using Viro’s approach. We give the complete KH table with .(a, b) bigrading, and we also convert it to the classical Khovanov (co)homology bigrading .(i, j). The process, as the reader may anticipate, is heavily based on the different states of a diagram D of the right-handed trefoil. Recall that for a projection with n crossings, there are .2n Kauffman states. In turn, for a state s with . |Ds | components, there are .2 |Ds | enhanced Kauffman states. First, we assign an ordering of the crossings. Next, we establish all eight Kauffman states of the trefoil. These states are obtained after considering all possible assignments of markers A and B and the smoothing of each crossing; see Fig. 19.4. The label under each state represents the markers assigned to each crossing according to the ordering. For instance, the label ABA means that the crossings 1 and 3 are assigned an A marker, while the crossing 2 is assigned a B marker.

Fig. 19.4: Kauffman states of the right-trefoil knot

19.4 Translation to Classical Khovanov (Co)homology

343

Having the Kauffman states, we are now able to find the enhanced Kauffman states of the trefoil. Recall that the grading .a = σ(s) depends only on the markers of the state. On the other hand, the value of the grading b depends additionally on the signs of the circles of the enhanced state. We find that the values .(a, b) of the bigrading that may yield nontrivial chain groups are the pairs (3,7), (3,3), (3,.−1), (1,7), (1,3), (1,.−1), (.−1,3), (.−1,.−1), (.−1,.−5), (.−3,3), (.−3,.−1), (.−3,.−5), and (.−3,.−9). For instance, Fig. 19.5 illustrates the enhanced Kauffman states corresponding to the state BBB and the obtained values for the gradings a and b.

Fig. 19.5: Enhanced Kauffman states for the state BBB With the calculation of the grading values .(a, b), the ground is set for the construction of the chain complex. Recall the definition of the boundary map . ∂a,b : Ca,b −→ Ca−2,b , and note that the value of b remains unchanged. Therefore, depending on this value, we construct our complex. For example, Fig. 19.6 shows all the enhanced Kauffman states giving the grading .b = −5.

Fig. 19.6: Enhanced states giving the grading .b = −5 By definition, .C−1,−5 = ZS−1,−5 where S−1,−5 = {S e EKS | σ(s) = −1, σ(s) + 2τ(S) = −5} .

.

Similarly, .C−3,−5 = ZS−3,−5 where S−3,−5 = {S e EKS | σ(s) = −3, σ(s) + 2τ(S) = −5} .

.

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19 Khovanov Homology: A Categorification of the Jones Polynomial

Thus, the following chain complex for the grading value .b = −5 is obtained; see Fig. 19.7. Pairs of incident states are also illustrated.

Fig. 19.7: Chain complex for .b = −5 ker(∂−1,−5 ) ker(∂−3,−5 ) is trivial. To calculate .H−3,−5 = , im(∂1,−5 ) im(∂−1,−5 ) observe that .ker(∂−3,−5 ) = C−3,−5 is the free abelian group generated by the following enhanced states (for convenience in the calculation, we denote them by . x, y, z) as illustrated in Fig. 19.8. Notice that .H−1,−5 =

Fig. 19.8: Generators of .ker(∂−3,−5 ) On the other hand, the image of .∂−1,−5 is the free abelian group with basis as illustrated in Fig. 19.9.

Fig. 19.9: Basis for .im(∂−1,−5 ) Therefore, the homology group .H−3,−5 is given by the quotient .

Z {x, y, z} . {x + z, −y − z, x + y}

19.5 Kauffman Bracket Polynomial as an Euler Characteristic of Khovanov Homology

345

The relations are equivalent to . z = −x = y and .2x = 0. Thus, the quotient is Z2 , a 2-torsion group. The process of computing the homology groups is analogous for other values of b (see Exercise 19.6.3). We have chosen .H−3,−5 to illustrate the fact that Khovanov homology can have torsion. The full Khovanov homology of the right-trefoil knot is given in Tables 19.3 and 19.4.

.

b | a

.

7 3 .−1 .−5 .−9

−3

.

−1

1

.

3 Z Z

. .

Z

.

Z2 .Z

.

Table 19.3: KH of the right-trefoil knot in Viro’s .(a, b) bigrading

j | i

0

.

9 7 5 3 1

1

2

3 .Z Z2

.

Z

.

Z .Z .

Table 19.4: KH of the right-trefoil knot in Khovanov’s original .(i, j) bigrading

19.5 Kauffman Bracket Polynomial as an Euler Characteristic of Khovanov Homology We think of a categorification of a numerical or polynomial invariant as a homology theory whose Euler characteristic is the given invariant. As we mentioned, Khovanov homology in Viro’s description is the categorification of the Kauffman bracket polynomial. By looking at it from a different perspective, we would like to obtain the Kauffman bracket polynomial as the (polynomial) Euler characteristic of the Khovanov homology. We will interpret the state sum formula E .[D] = (−1) |Ds | Aσ(s)+2τ(S), S e EKS

as a formula for the Euler characteristic; the important information we should take into account is that .∂a,b : Ca,b → Ca−2,b . We know from algebraic topology that

346

19 Khovanov Homology: A Categorification of the Jones Polynomial

the Euler characteristic of a chain complex can be computed from homology as well as from chain groups (boundary maps are not important as long as we know the degree of the differential map). In the classical case of E groups .Cn and differential . ∂n : Cn → Cn−1 , the Euler characteristic is defined by . (−1) j rankCj . However, j

sometimes, Ewe shift the chain complex, say by s, and then the Euler characteristic becomes . (−1) j+s rankCj . If .∂n : Cn → Cn−2 , then the Euler characteristic is j E j+s defined as . (−1) 2 rankCj , where j has the same parity as s. In our case, we deal j

with a bigraded chain complex .C = chain complex.

{(

)} Ca,b, ∂a,b , but for fixed b, we have a graded

For bigraded (finitely generated) groups .G a,b , theE Poincaré polynomial (i.e., generating function) is the two-variable polynomial . ca,b x a y b where .ca,b = a,b

rankG a,b . Of course, polynomials will be different for chain groups .Ca,b and the Khovanov homology groups .Ha,b . In the latter case, the Poincaré polynomial is an invariant of unoriented framed links, which catches the free part of the homology. For every fixed b, we can deal with the classical Euler characteristic so it can be computed from either chain groups or homology groups. To obtain the Euler characteristic, we substitute . x = −i and . y = i A in the Poincaré polynomial. In fact, we obtain not only the Euler characteristic but also the state sum formula for the unreduced Kauffman bracket polynomial: E E b−a ca,b (−i)a (i A)b = ca,b (−1) 2 Ab .

=

E

a,b

rankCa,b (−1)

a,b

b−a 2

A = b

E

(−1)

b−a 2

|Sa,b | Ab

a,b

a,b

=

E

(−1)

b−a 2

Ab .

S e EKS

Now, we observe that . b−a 2 = τ(S) ≡ |Ds | modulo 2 and our formula is exactly the Kauffman bracket state sum of unreduced Kauffman bracket polynomial. The fact that we substitute . x = −i and not .−1 is caused by the fact that the differential is going down by 2 at index a. The substitution . y = i A allows us to work with the integer . b−a 2 . For instance, for the right-handed trefoil knot, the Poincaré polynomial from the Khovanov homology is given as follows:

19.6 Exercises

347

E

ca,b x a y b = c−3,−9 x −3 y −9 + c−1,−1 x −1 y −1 + c3,3 x 3 y 3 + c3,7 x 3 y 7

a,b

= x −3 y −9 + x −1 y −1 + x 3 y 3 + x 3 y 7 x = −i, y = i A −−−−−−−−−−−−−→ [D] = (−i)−3 (i A)−9 + (−i)−1 (i A)−1 + (−i)3 (i A)3 + (−i)3 (i A)7 = −i −12 A−9 + −i −2 A−1 + −i 6 A3 + −i 10 A7

.

= −A−9 + A−1 + A3 + A7 A7 + A3 + A−1 − A−9 −A2 − A−2 5 −3 −7 = −A − A + A , =⇒ < D >=

which is well known to be the classic Kauffman bracket polynomial of the minimal diagram of the right-handed trefoil knot.

19.6 Exercises Exercise 19.6.1 Explain how the state sum formulas for both the reduced and unreduced KBP are obtained. Once you are certain that the formulas work, use them to find the KBP of the trefoil and the Hopf link. Exercise 19.6.2 Consider the diagram of the Hopf link with two crossings. Compute the KH of the Hopf link following Viro’s approach, and verify that the following Khovanov homology tables (see Table 19.5) are obtained (in the case of H i, j , assume that both crossings are positive). b | a 6 2 −2 −6

−2

Z Z

0

2 Z Z

j | i 6 4 2 0

0

1

2 Z Z

Z Z

Table 19.5: Khovanov homology tables for the Hopf link

Exercise 19.6.3 Complete the calculation of KH of the right-trefoil knot, and verify the answer given in Table 19.3. Exercise 19.6.4 Calculate the KH of the left-trefoil knot (mirror image of the righttrefoil knot), and compare with Exercise 19.6.3. ¯ = Exercise 19.6.5 Show that if D¯ is the mirror image of the diagram D, then Ha,b ( D) free(H−a,−b (D)) ⊕ tor(H−a−2,−b (D)).

Lecture 20 Long Exact Sequence of Khovanov Homology and Torsion

Khovanov homology is currently an active research area in knot theory in part, arguably, because it categorifies the Jones polynomial. In this lecture, we examine the fact that the Kauffman skein relation, used to define the Kauffman bracket polynomial, can be categorified by a long exact sequence. We use the long exact sequence to compute the Khovanov homology of the torus link of type .T (2, n). Moreover, we present one of the most elusive aspects of the theory, torsion in Khovanov homology. Aiming for a chronological showcase of the discoveries related to the torsion groups, we mention them in the order they were discovered. Only recently, infinite families of links with specific torsion orders were discovered, giving rise to a new wave of investigations.

20.1 Long Exact Sequence of Khovanov Homology In this section, we present a categorification of the Kauffman bracket skein relation following Viro [Vir1]. This categorification is given in the form of a long exact sequence of Khovanov homology. Recall from Definition 19.3.2 of Lecture 19 that the set of enhanced Kauffman states, .Sa,b = {S ∈ EKS | a = σ(s), b = σ(s) + 2τ(S)}, serve as a basis for the free abelian groups .Ca,b = ZSa,b . Since the long exact sequence of Khovanov homology is constructed for any fixed crossing v of a link diagram D, we define the A,v B,v and .Sa,b depending on the crossing v as follows: sets .Sa,b { } A,v Sa,b = S ∈ Sa,b | s(v) = A

.

and

{ } B,v Sa,b = S ∈ Sa,b | s(v) = B ,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5_20

349

350

20 Long Exact Sequence of Khovanov Homology and Torsion

A,v B,v and then it follows that .Sa,b = Sa,b u Sa,b . Following an analogous notation from the previous lecture, denote the free abelian groups generated by these sets as

.

where .v = equalities:

. In this way, on the level of groups, we have the following

.

Observe that the complex

( ) , is a chain subcomplex of . Ca,b, ∂a,b

, which is not necessarily true for i.e., since .∂a,b may change the marker at the crossing from an A marker to a B marker. Following the definitions and the construction so far, we can write the following short exact sequence of chain complexes:

.

where the map .α sends an EKS of D with the crossing v with a B marker to the EKS of D assigning a B label to v, while the other crossings keep the markers and the signs of the circles are preserved. Similarly, the map . β sends an EKS with B label at v to zero and sends each EKS with A label at v to the EKS of D with the crossing v given an A label, while the other markers of crossings and signs of circles are preserved. Notice that as a group necting map

, and there is a chain complex con-

.

Thus, in the standard way, this leads to a long exact sequence of homology:

20.1 Long Exact Sequence of Khovanov Homology

351

.

Finally, notice that we have the following chain complex equalities:

.

which follow because, in the former case, one B marker is lost after the smoothing and .τ is unchanged. Similarly, in the latter case, one A marker is lost after the smoothing, and .τ is unchanged. Hence, after using the isomorphisms, we obtain the following short exact sequence of chain complexes of diagrams:

.

which, in turn, yields the following long exact sequence (LES):

.

(20.1) The following result follows from the construction above. Corollary 20.1.1 (1) If phism. (2) If phism.

, then

, then

is a monomor-

is an epimor-

352

(3) If an isomorphism.

20 Long Exact Sequence of Khovanov Homology and Torsion

, then

is

Example 20.1.2 As an example, we will compute the Khovanov homology of the torus knots of type .T (2, n). This calculation was first performed by Khovanov, and a different proof was given by the first author by observing the connection of Khovanov homology and Hochschild homology [Kho1, Prz22]. The next definition introduces the notion of torus link of the type .(p, q). Definition 20.1.3 A torus link of type (p,q), also referred to as a .(p, q)-torus link, is a link ambient isotopic to a curve contained in a standard torus .T 2 . This curve wraps p times around the longitude, and it wraps q times around the meridian. If p and q are relatively prime, then the torus link has only one component, and therefore, it is called a torus knot. Notice that a vertical smoothing at a crossing v of .T (2, n) produces the trivial knot (with .1 − n twists). We write the long exact sequence of homology according to Eq. 20.1. Following our settings, keeping the crossing would yield .T (2, n), the horizontal smoothing would be .T (2, n − 1), and the vertical smoothing would be the trivial knot with framing twisted .(1 − n) times. Figure 20.1 illustrates the situation for the torus knot .T (2, 3).

Fig. 20.1: Torus knot .T (2, n) = T(2, 3) (center), torus link .T(2, n − 1) = T(2, 2) (left), and the trivial knot with framing .1 − n = −2 (right)

Theorem 20.1.4 Let .T (2, n) be a torus link of type .(2, n) with .n > 0. Then, Ha,b (T(2, n)) is given by

.

⎧ Z ⎪ ⎪ ⎪ ⎪Z ⎪ ⎨ ⎪ Z . Ha,b (T(2, n)) = ⎪ ⎪ Z2 ⎪ ⎪ ⎪ ⎪0 ⎩

for (a, b) = (n, n) or (a, b) = (−n, −3n), for a = n − 2s, b = n − 4s + 4 where s is even and 0 ≤ s ≤ n, for a = n − 2s, b = n − 4s where s is odd and 3 ≤ s ≤ n, for a = n − 2s, b = n − 4s + 4 where s is odd and 3 ≤ s ≤ n, otherwise.

Proof As we mentioned, the main tool used in proving this theorem is the long exact sequence of Khovanov homology. We now present a sketch of the proof. Parts of the proof can be generalized to alternating links and to adequate links and will be described later in the lecture.

20.1 Long Exact Sequence of Khovanov Homology

353

Our proof proceeds by induction on n. For the trivial knot .T (2, 1), the theorem holds as only .H1,1 (T(2, 1)) = H1,5 (T(2, 1)) = Z and the other homology groups are trivial. For the Hopf link .T (2, 2), the theorem holds as the reader can verify from Table 19.5 of Lecture 19. As the inductive hypothesis, suppose the theorem holds for .n − 1, where .n > 2. Now consider the case when the map . β∗ : Ha,b (T(2, n)) → Ha−1,b−1 (T(2, n − 1)) is not necessarily an isomorphism (see Corollary 20.1.1). As a result, we need to carefully analyze the long exact sequence of Khovanov homology of .T (2, n). First, notice that .TB (2, n) = O1−n , that is, it is the trivial knot with framing .1 − n. Thus, its homology is given by { Z when (x, y) = (1 − n, 3(1 − n) ± 2), 1−n . Hx,y (O )= 0 otherwise. Therefore, the map . β∗ is not necessarily an isomorphism if .Hx,y (O1−n ) = 0. More precisely, .Hx,y (O1−n ) = 0 when (x, y) = (1 − n, 3(1 − n) − 2) = (1 − n, 1 − 3n) or

.

(x, y) = (1 − n, 3(1 − n) + 2)) = (1 − n, 5 − 3n).

.

(1) Let .(x, y) = (1 − n, 1 − 3n). In the long exact sequence, the neighborhood of 1−n ) has the form . H1−n,1−3n (O α∗ −−−−−−−→ H1−n,−1−3n (T(2, n − 1)) −−−−−−−→ H1−n,1−3n (O1−n ) −−−−−−−→ .

β∗ H−n,−3n (T(2, n)) −−−−−−−→ H−n−1,−3n−1 (T(2, n − 1)) −−−−−−−→ .

By the inductive hypothesis, the homology groups .H1−n,−1−3n (T(2, n − 1)) and H−n−1,−3n−1 (T(2, n − 1)) are trivial. Hence, the previous long exact sequence becomes ( Conn ) ∂ α∗ −−−−−−−→ 0 −−−−−−−−−−−−−→ H1−n,1−3n (O1−n ) −−−−−−−−−−−−→

.

.

β∗ H−n,−3n (T(2, n)) −−−−−−−−−−→0 −−−−−−−→ .

Thus, .H−n,−3n (T(2, n)) = H1−n,1−3n (O1−n ) = Z, as needed. (2) Let .(x, y) = (1 − n, 5 − 3n). In the long exact sequence, the neighborhood of 1−n ) has the form . H1−n,5−3n (O

354

20 Long Exact Sequence of Khovanov Homology and Torsion

) Conn ∂ 1−n,3−3n β∗ ∗ 0 −−−−−−−→ H2−n,4−3n (T(2, n)) −−−−−−−→ H1−n,3−3n (T(2, n − 1)) −−−−−−−−−−→ (

β∗ α∗ H1−n,5−3n (O1−n ) −−−−−−−→ H−n,4−3n (T(2, n)) −−−−−−−−−−→ H−n−1,3−3n (T(2, n − 1)) −−−−−−−→ .

.

Similarly as before, by the inductive hypothesis, .H−n−1,3−3n (T(2, n − 1)) = 0 and .H1−n,3−3n (T(2, n − 1)) = H−(n−1),−3(n−1) = Z. Thus, the previous long exact sequence becomes

.

( Conn ) ∂ β∗ 0 −−→ H2−n,4−3n (T(2, n)) −−−−−−−→ Z −−−−−−−−−→ β∗ α∗ Z −−−−−−−→ H−n,4−3n (T(2, n)) −−−−−→ 0 −−→ .

In this way, to find the other entries of the homology in this part of the sequence, Conn : Z → Z. We consider two general we need to understand the map .∂1−n,3−3n possibilities: Conn (i) Suppose .∂1−n,3−3n is the zero map.

In this case, we have .H2−n,4−3n (T(2, n)) = H1−n,3−3n (T(2, n − 1)) = Z and H−n,4−3n (T(2, n)) = H1−n,5−3n (O1−n ) = Z. Then, .H2−n,4−3n (T(2, n)) = Z, and in this case, .(a, b) = (2 − n, 4 − 3n) = (n − 2n + 2, n − 4n + 4) = (n − 2(n − 1), n − 4(n − 1)) = (n − 2s, n − 4s) for .s = n − 1. Similarly, . H−n,4−3n (T(2, n)) = Z where .(a, b) = (−n, 4 − 3n) = (n − 2s, n − 4s + 4) for . s = n and the theorem holds. .

Conn is a multiplication by . k > 0. (ii) Suppose .∂1−n,3−3n

In this case, .H2−n,4−3n (T(2, n)) = 0 and .H−n,4−3n (T(2, n)) = Zk . To complete the proof of the theorem, we should check what is the value of k. This case was addressed in 2006 by Pabiniak, Sazdanović, and the first author in [PPS] by studying the algebra of truncated polynomials .A m = Z[x] / (x m ), which for .m = 2 is closely related to Khovanov homology as previously described. The checking is a direct calculation depending on whether n is even or odd. We leave it to the reader. More precisely, the following cases are considered: Conn (i’) For even n, we are in the case where .∂1−n,3−3n is the zero map, and we have .H2−n,4−3n (T(2, n)) = Z = H−n,4−3n (T(2, n)).

20.1 Long Exact Sequence of Khovanov Homology

355

Conn (ii’) For odd n, we are in the case where .∂1−n,3−3n is a multiplication by . k > 0 with . k = 2, and we get .H2−n,4−3n (T(2, n)) = 0 and .H−n,4−3n (T(2, n)) = Z2 .

u n We encourage the reader to study the previous proof by looking at Tables 20.1 and 20.2 where regularity of homology is visible. Later in the lecture, it will be seen that similar results hold for adequate diagrams [PPS]. b | a .−11 .−9 .−7 .−5 .−3 .−1 1 3 5 7 9 11 .Z 15 .Z 11 .Z 7 .Z2 3 .Z .Z .−1 .Z2 .−5 .Z .Z .−9 .−13 .Z2 .−17 .Z .Z .Z2 .−21 .−25 .Z .Z .−29 .Z2 .Z .−33

.

Table 20.1: Khovanov homology of the torus knot .T(2, 11) Notice that .Ha,b (T(2, n)) has support on slope 2 diagonals containing .(n, n) or (n, n + 4). That is, .Ha,b (T(2, n)) is nontrivial only for .Hn−2s,n−4s or .Hn−2s,n−4s+4 . We see also that .Ha,b (T(2, n)) has torsion only in groups .Hn−2s,n−4s+4 (see tables). Both observations can be generalized to alternating links.

.

Motivated by the previous example, we now generalize the ingredients used to compute the homology of torus knots of the type .T (2, n). In fact, the method used in the previous proof can be also used to construct many links whose Khovanov homology has torsion different from .Z2 . A few related results of this type will be described later in this lecture [MPSWY]. Denote by .s A the Kauffman state which has an A-marker at all crossings; analogously, denote by .sB the Kauffman state that has a B-marker at all crossings. The notion of adequacy is a natural generalization of the alternating property of diagrams without nugatory crossings (i.e., a crossing as in Fig. 20.2a) which is suitable for the study of Khovanov homology. We define adequate diagrams using properties of the states .s A and .sB . Definition 20.1.5 A link diagram D is said to be A-adequate (respectively, Badequate) if the state of positive markers .s A (respectively, .sB ) cuts the diagram to

356

20 Long Exact Sequence of Khovanov Homology and Torsion

b | a .−12 .−10 .−8 .−6 .−4 .−2 0 2 4 6 8 10 12 .Z 16 .Z 12 .Z 8 4 .Z2 .Z .Z 0 .Z2 .−4 .−8 .Z .Z .Z2 .−12 .−16 .Z .Z .−20 .Z2 .−24 .Z .Z .Z2 .−28 .Z .Z .−32 .−36 .Z

.

Table 20.2: Khovanov homology of the torus link .T(2, 12)

the collection of circles so that every crossing connects different circles. The diagram D is said to be adequate if the diagram is both A- and B-adequate. Figure 20.2b illustrates that the minimal crossing diagram of the right-trefoil knot is adequate.

Fig. 20.2: Generalization of the alternating property of a knot diagram Recall from Lecture 2 that Tait was the first to notice the connection between knots and planar graphs. Tait’s construction of a planar graph from an alternating diagram is generalized to the notion of an associated graph as follows. We also introduce some notation that will be useful in the statements of the main results presented in the lecture; compare Lecture 5. Definition 20.1.6 Let D be a link diagram and s a Kauffman state of D, and consider the states .s A and .sB as before. 1. The associated graph to D and the state s, .G s (D), is constructed as follows. Vertices of .G s (D) correspond to circles of . Ds . Edges of .G s (D) are in bijection with crossings of D, and an edge connects two vertices if the corresponding crossing connects the circles of . Ds corresponding to the vertices.

20.1 Long Exact Sequence of Khovanov Homology

357

2. In the language of associated graphs, we can state the definition of adequate diagrams as follows: the diagram D is A-adequate (resp., B-adequate) if the graph .G s A (D) (resp., .G sB (D)) has no loops. 3. The girth of a state s, .l(s), is the girth of the graph .G s (D), i.e., the length of the shortest cycle in .G s (D) (in the case where G is a forest, by convention, .l(s) = ∞). The diagram D is said to be s-adequate if .l(s) > 1. The diagram D is said to be strongly adequate if .l(s) > 2. Figure 20.3 shows an illustration of the process of constructing the associated graphs .G s A and .G sB for a minimal diagram of the Whitehead link (leftmost diagram in the picture).

Fig. 20.3: Minimal diagram of the Whitehead link and graphs associated with the states .s A and .sB From the construction of Khovanov homology presented in Lecture 19 and some properties of the Kauffman states, we can deduce the following statements. Let D be a link diagram of n crossings, and consider the states .s A and .sB as defined above. Denote the number of circles in the states .s A and .sB by . |s A | and . |sB |, respectively. Then, the highest (or outermost) term of the Khovanov chain complex is given by .Cn,n+2 |s A | (D) = Z. Furthermore, if the diagram D is A-adequate, then the whole group .C∗,n+2 |s A | (D) = Z and .Hn,n+2 |s A | (D) = Z. Analogously, the lowest term in the Khovanov chain complex is given by .C−n,−n−2 |sB | (D) = Z. Additionally, if the diagram D is B-adequate, then the whole group .C∗,−n−2 |sB | (D) = Z and . Hn,−n−2 |s B | (D) = Z. Exercise 20.1.7 Show that if D is an alternating diagram, then .G s A and .G sB are the plane graphs first constructed by Tait. Exercise 20.1.8 Show that the following equalities hold for the Khovanov homology of an A- or B-adequate diagram. .

Hcr(D),cr(D)+2 |Ds A |) = Z for A-adequate diagram.

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20 Long Exact Sequence of Khovanov Homology and Torsion .

H−cr(D),−cr(D)−2 |Ds B |) = Z for B-adequate diagram.

Exercise 20.1.9 Show that an alternating diagram with no nugatory crossing is Aand B-adequate. Exercise 20.1.10 Let D be a connected alternating link diagram. Then the Khovanov homology of D is supported on two adjacent diagonals of slope 2, containing the two points .(cr(D), cr(D) + 2|Ds A |) and .(−cr(D), −cr(D) − 2|DsB |).

In the next exercise, we define and compute the twisted homology of a graph G. For standard homology of a graph with the definition of boundary instead of difference, we take the sum (if e is an edge with endpoints .v1 andv2 (possibly .v1 = v2 ), we define . ∂1 (e) = v1 + v2 ). We start from the graph .(V, E) and define .C0 = ZV and .C1 = ZE. Let us denote the homology of the above chain complex by .Hi(t) (G). Exercise 20.1.11 Let .G = (V, E) be a graph. The twisted graph homology of G, (t) . H (G) (.i = 0, 1), is defined as follows: .C0 = ZV, .C1 = ZE, and . ∂1 (e) = v1 + v2 for i .v1 and .v2 endpoints of e not necessarily different. Show that for a connected graph G, we have (t) . H (G) 0

{ = ZV/(v1 (e) + v2 (e)) =

(t) . H (G) 1

{ = ker∂1 =

Z when G is bipartite, Z2 when G has an odd cycle,

Z |E |− |V |+1 when G is bipartite, Z |E |− |V | when G has an odd cycle.

Exercise 20.1.12 If D is a connected B-adequate with associated graph not bipartite (e.g., .(2, n) knot), then .

H−cr(D),−cr(D)−2 |Ds B |+4 = Z2 .

Hint: Use Exercise 20.1.11.

20.2 Torsion in Khovanov Homology A powerful link invariant, Khovanov homology has been computed for many links, and the experimental evidence shows abundance of 2-torsion, while other torsion groups appear very seldom; see, for instance, [Kho1, Shu1]. We now present an overview of the developments and the discoveries related to the torsion groups in the Khovanov homology theory.

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Alexander Shumakovitch verified that for prime knots up to 14 crossings only Z2 -torsion appears and up to 16 crossings the only other torsion that appears is .Z4 -torsion [Shu1]. In fact, 38 knots with 15 crossings (one of them being the .T(4, 5) torus knot) and 129 knots with 16 crossings have .Z4 -torsion. In other words, very few knots up to 16 crossings contain .Z4 -torsion in its KH, and there is no knot with .Z3 -torsion or with larger-order torsion. The reader who attempted Exercise 3 in Lecture 19 may realize that the calculation of KH gets difficult to do by hand, especially as the number of crossings increases. Consequently, the use of computer programs to aid in the calculations is in place. Dror Bar-Natan developed a Mathematica software package for KH calculations. In 2002, Shumakovitch created the program KhoHo for calculating KH, emphasizing that it works particularly well for links with up to 17–19 crossings [Shu2]. KhoHo can be used to compute even Khovanov homology (the original version of KH) [Kho1], odd KH [ORS] (since 2008), and the unified version of KH [Put] (since 2018). In 2005, Dror Bar-Natan and J. Green designed another program, called JavaKh1 in [BG] based on the algorithm introduced in [BarN3]. The algorithm allowed him to detect odd torsion in Khovanov homology: .Z3 - and .Z5 -torsion appear in the KH of the torus knot .T (5, 6), and .Z7 -torsion appears in the KH of the torus knot .T (7, 8). Following this algorithm, in 2014, Lukas Lewark computed the KH of the torus knot .T (8, 9) and found .Z8 -torsion. Using the long exact sequence of Khovanov homology, in 2016, the authors of [MPSWY]2 constructed infinity families of knots and links containing .Z3 -, .Z5 -, and .Z7 -torsion in their Khovanov homology. They also showed the existence of .Zn -torsion for .2 < n < 9 and .Z2 s -torsion for . s ≤ 23 in the Khovanov homology of some family of knots and links with braid index 4. In early 2019, Sujoy Mukherjee introduced the first known examples of knots and links with torsion of orders 9, 25, 27, and 81 [Muk]. Recently, Mukherjee and Dirk Schütz constructed links with arbitrarily large torsion [MuSc]. .

2-Torsion in Khovanov Homology Torsion in Khovanov homology is an area of active research. One of the main goals is to interpret the torsion information in terms of topological properties of the link. In 2003, Shumakovitch stated the following Conjecture 20.2.1, which, if proved, would yield a way of detecting the unknot. The first author together with Marta M. Asaeda in [AP] partially solved Shumakovitch’s conjecture by proving it for some adequate diagrams. Conjecture 20.2.1 (Shumakovitch) The Khovanov homology of every link, except the unknot, the Hopf link, their disjoint unions, and their connected sums, has torsion of order 2. The authors of [AP] used the notion of adequacy of diagrams; see Definition 20.1.5. As previously mentioned, they noted that for an n-crossing, Aadequate link diagram D, the highest grading in Khovanov homology is given by 1 Now known as JavaKh-v1. An update of this program, JavaKh-v2, was written by Scott Morrison. 2 This paper results from Mathathon II which took place December 13–23, 2016.

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Hn,∗ (D) = Hn,n+2 |s A | (D) = Z. Moreover, they showed that the next nontrivial homology group, .Hn−2,n+2 |s A |−4 (D), has .Z2 -torsion as long as the graph associated with the state .s A, .G(D) = G s A (D), is not bipartite. Furthermore, the group .Hn−2,n+2 |s A |−4 (D) was explicitly computed for A-adequate links in [PPS]. In particular, for a connected A-adequate diagram D, { ( ) Z2, f or G(D) having and odd cycle .tor Hn−2,n+2 |s A |−4 (D) = 0, f or a bipartite graph.

.

Asaeda and the first author went a step deeper and analyzed the torsion in Hn−4,n+2 |s A |−8 (D), the next nontrivial homology group. They showed that for a strongly A-adequate diagram D with the graph .G(D) containing an even cycle, this group contains .Z2 -torsion. This statement implies Shumakovitch’s result that any alternating link which is not a connected or disjoint sum of trivial links and Hopf links has a nontrivial .Z2 -torsion in its Khovanov homology [Shu1].

.

Finally, in [PrSa], the first author and Radmila Sazdanović computed the entire homology group .Hn−4,n+2 |s A |−8 (D) for many classes of A-adequate diagrams, including strongly A-adequate diagrams. In particular, for a connected n-crossing A-adequate diagram, D, ( p (G' (D))−1 ( ) Z2 1 , f or G '(D) having an odd cycle; .tor Hn−4,n+2 |s A |−8 (D) = p1 (G' (D)) Z2 , f or G '(D) a bipartite graph, where .G(D) = G s A (D) and .G '(D) is a simple graph obtained from .G(D) by replacing multiple edges by singular edges and . p1 is the cyclomatic number . p1 (G ') = |E |−v+p0 where . p0 denotes the number of components (in our case, . p0 = 1). Torsion of Other Orders In 2017, Sujoy Mukherjee, Marithania Silvero, Xiao Wang, Seung Yeop Yang, and the first author published the paper Search for torsion in Khovanov homology, [MPSWY]. The article contains an abundance of information on torsion in Khovanov homology. In several cases, examples of knots smaller than the previously known ones, containing a given torsion group, are exhibited. The main families of links analyzed in the paper are twist deformations of torus links. A torus link .T (m, n) with .m ≥ 2 can be represented as the closed braid on m strands . βˆm,n = (σm−1 σm−2 · · · σ2 σ1 )n , with the .σi ’s being the classic generators of the braid group given by E. Artin [Art1]; see Lecture 8. For . k ∈ Z, denote by (k) (m, n) the link represented by the closed braid . βˆ n k .T m,n = (σm−1 σm−2 · · · σ2 σ1 ) σ1 , i.e., the link .T (m, n), after adding k half-twists on the first two strands.3 Figure 20.4 illustrates .tk -moves, the torus link .T (3, 5), and the twist deformation .T (2) (3, 5). 3 When .k > 0, we add k positive half-twists, and when .k < 0, we add k negative half-twists.

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Fig. 20.4: .tk -move (left), torus links .T (3, 5) (center), and .T (2) (3, 5) (right)

Aiming to illustrate the progress made in the investigation of torsion in Khovanov homology, we present a summary of the main discoveries made in [MPSWY]. A reader interested in a detailed discussion of the matter is invited to see [MPSWY, MuSc]. Before this paper, the smallest known knot with .Z3 -torsion was the 24-crossing torus knot .T (5, 6). Their computations show that the closure of the braid .(σ3 σ2 σ1 σ4 σ3 )4 , a 3-component link with at most 20 crossings, also exhibits .Z3 -torsion in its KH. They also showed that the knot obtained by closing the 5-braid 4 −1 −1 .(σ3 σ2 σ1 σ4 σ3 ) (σ 4 σ2 ) has at most 22 crossings and also has .Z3 -torsion. The 15-crossing torus knot .T (4, 5) is the smallest known prime knot with .Z4 torsion; see [Shu1]. Another example of a knot with 15 crossings having .Z4 -torsion is the twisted torus knot .T (−3) (4, 6) [MPSWY], which is the 2-cabling of the trefoil with 3-framing. In [MPSWY], the authors showed that the closure of the braid 7 −5 σ −2 , obtained from the knot .T (−5) (4, 7) by adding two negative half.(σ3 σ2 σ1 ) σ 3 1 twists on the rightmost pair of strands, is a 2-component link. After reduction, this is the closure of the braid .σ2 σ12 (σ3 σ2 )2 σ1 σ3 σ22 σ1 σ3 σ2 , and it has .Z4 -torsion. This link of two components has at most 14 crossings. The smallest known knot with .Z5 -torsion was the 24-crossing torus knot .T (5, 6). The authors of [MPSWY] showed that the knot .T (−6) (5, 7) with 22 crossings has .Z5 torsion in its Khovanov homology. Additionally, they showed that the 2-component link .T (−7) (5, 7) with 21 crossings also exhibits .Z5 -torsion. Bar-Natan showed that the 48-crossing torus knot .T (7, 8) has .Z7 -torsion, and this was the smallest known such knot. In Search for torsion in Khovanov homology, they showed that the knot .T (−8) (7, 9) and the 2-component link .T (−9) (7, 9) with at most 46 and 45 crossings after reduction, respectively, contain .Z7 -torsion. Finally, the smallest known knot with .Z8 -torsion was the 63-crossing torus knot .T (8, 9). The Khovanov homology of the 2-component link .T (−8) (6, 8), with braid index 6, also has .Z8 -torsion. This link is equivalent to the closure of the braid word .(σ4 σ3 σ2 σ1 σ5 σ4 σ3 σ2 )4 which has 32 crossings. It is also equivalent to the 2-cabling of the .T (3, 4) torus knot (the knot .819 , first nonalternating knot). Moreover, they showed that the knot .T (−9) (6, 8), obtained from adding a half-twist to .T (−8) (6, 8), with at most 33 crossings, also ex-

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20 Long Exact Sequence of Khovanov Homology and Torsion

hibits .Z8 -torsion. At the moment of writing Search for torsion in Khovanov homology, there was no knot or link known to contain torsion larger than .Z8 . They improved this by constructing .Z2s -torsion, with .s ≤ 23, in the following result. Theorem 20.2.2 Suppose that .H 6s+1,16s+4 (T (−4s−2) (4, 4s +2)) has .Z2s -torsion where . s ≥ 1. Then, for any . k ≤ −4s − 2, the Khovanov homology .

H 6s+1,20s+6+k (T (k) (4, 4s + 2))

also contains .Z2s -torsion. Moreover, they stated the following conjecture, verified up to .s = 23. Conjecture 20.2.3 The flat 2-cabling of the torus knot .T (2, 2s + 1) has .Z2s' -torsion for .0 < s ' ≤ s. 1. in bidegree .(i, j) for .i = 1 + 8s − 2s ' and . j = 4 + 20s − 4s ', where s is a positive integer. 2. in bidegree .(i, j) for .i = 8s − 2s ' and . j = 20s − 4s ', where s is a positive integer greater than one. Before [Muk], the highest known odd torsion group present in KH was .Z7 . The next odd torsion group, .Z9 , was conjectured to appear in the KH of the .T(9, 10) torus knot, which has 80 crossings. It was too large and, consequently, was beyond computational reach.4 Mukherjee took on the challenge of finding or constructing a smaller knot or link exhibiting .Z9 -torsion [Muk]. Remarkably, besides accomplishing the initial task, he also found knots and links with .Z27 -, .Z81 -, and .Z25 -torsion in their KH. Denote by . Bn the braid group with .(n − 1) generators. Let .0 ≤ i = j < n. For .i < j, denote by .wi, j the braid word .σi σi+1 · · · σj σj σj−1 · · · σi . Theorem 20.2.4 (Mukherjee) 5 1. The closure of .(σ1 σ2 σ3 σ4 )5 · w1,4 ∈ B5 contains .Z9 -torsion in its KH. Also, the connected sum of the torus knot .T (5, 6) with itself contains .Z9 -torsion in its KH.

2. The closure of the braid .(σ1 σ2 σ3 σ4 )6 (σ4 σ5 σ6 σ7 )6 (σ7 σ8 σ9 σ10 )6 , the overlapping connected sum of the torus knot .T (5, 6) with itself twice, contains .Z27 torsion in its KH. 3. The connected sum of the closure of the braid .(σ1 σ2 σ3 σ4 σ5 )6 σ1 σ2 σ3 σ4 with itself contains .Z25 -torsion in its KH. 4 Dirk Schütz recently partially computed the KH of the .T (9, 10) torus knot, and he detected .Z9 and .Z27 -torsion [Schü].

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Similar to Conjecture 20.2.3, the following conjecture, verified up to .m = 4, proposes a family of links with torsion of order .3m in their KH. Conjecture 20.2.5 (Mukherjee) The KH of the link .T (2, 3)#m (σ1 σ2 σ3 )4 σ1 σ2 , for + .m ∈ Z , contains the torsion subgroups .Z3, Z9, . . . Z3 m , where .# denotes the connected sum operation. In Conjecture 20.2.5, the torus knot .T (2, 3) can be replaced with a copy of (σ1 σ2 σ3 )4 σ1 σ2 , but the trefoil is preferred as it has a lower crossing number. To conclude our overview of the torsion in Khovanov homology, we highlight the new results published by Mukherjee and Dirk Schütz. They constructed the first family of links with arbitrarily large torsion groups. For every positive integer k and . p ∈ {3, 5, 7}, they describe a link with .Z p k in its Khovanov homology; for details, see [MuSc]. .

Appendix A

Basics of Three-Dimensional Topology

This appendix serves as a means to familiarize the reader with some of the key concepts in the deep and rich theory of three-dimensional manifolds used in this book. We discuss manifolds from several category theoretic viewpoints—topological, differentiable, and piecewise linear. We also discuss handle decomposition and surgery in this general setting. Finally, we examine the structure of three-dimensional manifolds, including Heegaard decomposition, prime decomposition, and Jaco-ShalenJohansson decomposition. We also give concrete descriptions of lens spaces and Seifert fibered manifolds.

A.1 Introduction The subject of topology in its modern form was invented at the end of the nineteenth century by Henri Poincaré. In his groundbreaking paper from 1895 [Poi1] and a set of supplementary papers, which he published over the next few years, Poincaré laid out the foundations of the subject. The next few decades were marked by a rapid development in point-set topology. The first modern definition of a manifold based on point-set topology appeared in the paper [VW] by Oswald Veblen and John Henry Constantine Whitehead in 1931. The idea of a manifold predates Poincaré, however. In the 1820s, Carl Friedrich Gauss first used the concept of local coordinates on a surface and, hence, already worked with the notion of charts. In fact, he is credited to be the first to think of surfaces as abstract spaces in their own right, independent of being embedded into the Euclidean space. In 1854, Bernhard Riemann gave a lecture at Göttingen titled “Über die Hypothesen, welche der Geometrie zu Grunde liegen”, loosely translating to “On the hypotheses underlying geometry”, in which he introduced the notion of a “Mannigfaltigkeit”. This German word translates to the words “diversity” or “manifold” in English.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5

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This appendix serves as a means to acquaint the reader with some of the key concepts in the deep and extensive theory of three-dimensional manifolds used in this book. For a more comprehensive and thorough reading, we refer the reader to [Hem, Jac, Hat2, Mart, RoSa], and [Hud].

A.2 Definitions and Examples Definition A.2.1 A map f : X −→ Y between two topological spaces is called a homeomorphism if it is a bijection and bicontinuous, that is, f and f −1 are both continuous. Definition A.2.2 (Topological Manifolds) A topological manifold of dimension n, also known as an n-manifold, is a second countable Hausdorff topological space M for which there exists an indexed family of pairs {(Uα, φα )}α∈Λ that satisfy the following conditions: • ∀α, Uα is an open subset of M and M =

 α

Uα .

• ∀α, φα is a homeomorphism from Uα onto an open subset Vα of Rn . The pair (Uα, φα ) is often called a coordinate chart, or simply a chart, of the topological manifold M. The indexed family {(Uα, φα )}α∈Λ that covers M is called an atlas for M. Remark A.2.3 Many authors replace the requirement of second countability with the weaker condition of paracompactness, which is equivalent to the property that a manifold is metrizable. Indeed, every second countable topological space X is paracompact; however, the converse is true if and only if X has countably many connected components. Thus, a topological n-manifold is a second countable Hausdorff topological space that is locally homeomorphic to the Euclidean n-space, given by an atlas of homeomorphisms. Any two topological n-manifolds are equivalent if they are homeomorphic. Definition A.2.4 Let (Uα, φα ) and (Uβ, φβ ) be two charts for M such that Uα ∩ Uβ  . A transition map τα,β : φα (Uα ∩ Uβ ) −→ φβ (Uα ∩ Uβ ) is defined as τα,β = φβ ◦ φ−1 α . Notice that τα,β is a homeomorphism since φα and φβ are homeomorphisms. Thus, transition maps give us a way to compare any two charts of an atlas of M. See Fig. A.1.

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=

R

◦

−1

( (

)

) ◦

−1

Fig. A.1: A transition map between two charts

Remark A.2.5 When we add some additional structure to topological manifolds, we usually do this by adding some restrictions on the transition maps (see, e.g., Definition A.3.1). Example A.2.6 1. The Euclidean n-space Rn is an immediate example of a topological n-manifold, as is any open subset of Rn . The n-ball D n = {x ∈ Rn : ||x|| ≤ 1} is also a topological manifold. Note that every subset of Rn is second countable and Hausdorff. Hence, to show that a subset S of Rn is an n-manifold, we just need to show that every point in S has a neighborhood homeomorphic to Rn .  2 xi = 1} is an n-manifold. The 2. The n-sphere S n = {(x0, x1, . . . , xn ) ∈ Rn+1 | homeomorphism is given by stereographic projection, h : S n \{(0, 0, . . . , 1)} −→ Rn . Hence, any point x  (0, 0, . . . , 1) in S n has S n \ {(0, 0, . . . , 1)} as the neighborhood that is homeomorphic to Rn . To get a neighborhood of (0, 0, . . . , 1) that is homeomorphic to Rn , take the composition of h with the reflection in Rn × {0}, to obtain the homeomorphism h  : S n \ {(0, 0, . . . , −1)} −→ Rn . Thus, S n \ {(0, 0, . . . , −1)} is a neighborhood of (0, 0, . . . , 1) that is homeomorphic to Rn . Hence, {(S n \ {(0, 0, . . . , 1)}, h), (S n \ {(0, 0, . . . , −1)}, h )} forms an atlas for S n . 3. Let M and N be manifolds of dimensions m and n, respectively. If {(Gα, φα )} is an atlas for M and {(Hβ, ψβ )} is an atlas for N, then the family of charts {(Gα × Hβ ), φα × ψβ : Gα × Hβ −→ Rm × Rn } is an atlas for M × N. Thus, M × N is an (m + n)-manifold, also known as a product manifold. In fact, the n-fold Cartesian product of manifolds of dimensions mi is a manifold of

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dimension

n  i=1

mi . The n-torus T n = S 1 × S 1 × . . . × S 1 (n copies) is one such

example of an n-manifold. One can also obtain T n as the quotient space of Rn as follows. Let T be the group of translations of length one along the coordinate axes of Rn . Then, T n = Rn /∼, where x ∼ y ⇐⇒ t(x) = y for t ∈ T. 4. The n-dimensional real projective space RP n , which is a quotient space of S n and is obtained by identifying antipodal points on S n , is an n-manifold.1 Exercise A.2.7 1. Show that T n and RP n are n-manifolds. 2. Try to find the minimal atlases for S n , T n , and RP n (by minimal atlas, we mean the minimal number of charts in the atlas). Definition A.2.8 (Submanifolds) Let M be an n-manifold, K ⊂ M, and k ≤ n. Then K is said to be a k-dimensional submanifold of M if there exists an atlas {(Uα, φα )}α∈Λ of M such that for every x ∈ K, there is a chart (Uα, φα ) with x ∈ Uα and φα (K ∩ Uα ) = Vα ∩ Rk , Vα open in K. Note that K is a manifold in its own right. The codimension of a submanifold K of M is the difference n − k in the dimensions of M and K, and K is said to be a codimension n − k submanifold of M. Example A.2.9 The n-sphere S n is a codimension 1 submanifold of Rn+1 . Example A.2.10 Let f : Rn −→ Rm have continuous partial derivatives of all orders. Then, the graph of f defined as G = {(x1, x2, . . . , xn, y1, y2, . . . , ym ) ∈ Rn+m | f (x1, x2, . . . , xn ) = (y1, y2, . . . , ym )} is an n-dimensional submanifold of Rm+n . Definition A.2.11 An injective map f : M −→ N between two manifolds is called an embedding if its image f (M) is homeomorphic to M and f (M) is a submanifold of N. In this case, f (M) is said to be an embedded submanifold of N. However, one can also consider immersed submanifolds, in which case the map f is not injective. This implies that the image f (M) has self-intersections. For example, the Klein bottle is an immersed submanifold of S 3 . Definition A.2.12 (Manifolds with Boundary) An n-manifold with boundary is a second countable Hausdorff space M with an atlas {(Uα, φα )}α∈Λ such that for each α, φα is a homeomorphism from Uα to either 1 The skein modules of the manifolds in Example A.2.6, when n = 3, are discussed in Lectures 11 and 12.

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Rn or R+n = {(x1, x2, . . . , xn ) ∈ Rn | x1 ≥ 0}. The boundary of M, denoted by ∂ M, is the set of all points whose neighborhoods are homeomorphic to R+n and not to Rn . The interior of M, denoted by int(M), is the set of points whose neighborhoods are homeomorphic to Rn . Note that ∂ M is an (n − 1)-manifold contained in M; however, it is not a submanifold of M. Example A.2.13 Consider an oriented 2-manifold, also known as a surface, with genus g and b number of D2 interiors removed. Denote this manifold by Fg,b . Then Fg,b is a surface with boundary if b ≥ 1. Thus, the disk, D2 = F0,1 ; pair of pants, F0,3 ; torus with a hole, F1,1 ; and annulus, F0,2 , are examples of 2-manifolds with boundary. The Möbius band, Mb, is also a surface, albeit non-orientable, with boundary. Example A.2.14 F0,b × [0, 1], b ≥ 1 are examples of 3-manifolds with boundary. Such manifolds are called handlebodies. Their Kauffman bracket skein modules and algebras are discussed in detail in Lecture 13. Note that the boundary of the handlebody F0,b × [0, 1], b ≥ 1, is the surface Fb−1,0 . See Definition A.4.7. Example A.2.15 The product of the Möbius band with the unit interval, also known as a solid Klein bottle, is a 3-manifold whose boundary is the Klein bottle. Example A.2.16 The n-ball D n is an example of an n-manifold with boundary and ∂D n = S n−1 . Definition A.2.17 An n-manifold M is said to be compact if it is compact as a topological space. If ∂ M = ∅ and M is compact, then M is said to be closed. If ∂ M = ∅ and M is noncompact, then M is said to be open. Example A.2.18 The 2-sphere, torus, real projective plane, and Klein bottle are closed 2-manifolds. In fact, S n , T n , and RP n are all closed n-manifolds. We can divide topological manifolds into two disjoint classes: orientable and non-orientable. Definition A.2.19 A topological n-manifold is said to be orientable if it has no generalized Möbius band embedded in it. The generalized Möbius band is the Cartesian product M b × D n−2 . Moreover, compact connected orientable topological n-manifolds can be characterized by Hn (M, ∂ M) = Z. An orientation of these manifolds is a choice of a generator for Z.2 A manifold together with a choice of orientation is said to be oriented. A homeomorphism between manifolds is said to be orientation preserving if the induced homomorphism on the nth homology groups maps 1 → 1 and orientation reversing if it maps 1 → −1. 2 For example, Hn (R P n ) = Z, when n is odd, and Hn (R P n ) = Z2 , when n is even. Thus, R P n is orientable when n is odd and unorientable when n is even.

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(a) Möbius band.

(b) Klein bottle.

(c) Torus.

Fig. A.2: The Möbius band, Klein bottle, and torus can be obtained by gluing the edges of the rectangles according to the arrow labelings

Exercise A.2.20 Every closed surface can be obtained from an oriented polygon with an even number of sides by pairwise identification of the sides. Such polygons are known as fundamental polygons of the surfaces they represent. See Fig. A.2 for an illustration. Draw the fundamental polygons of S 2 and RP2 . Example A.2.21 Rn , S n , and T n are orientable n-manifolds for all n. RP n is orientable for every odd n and is unorientable otherwise. Example A.2.22 The Möbius band, Klein bottle, and real projective plane are nonorientable surfaces. Example A.2.23 The Cartesian product of orientable n-manifolds is orientable. Thus, the product of an orientable surface with the unit interval I is an orientable 3manifold. The twisted product of a non-orientable surface with I is also an orientable 3-manifold. The majority of the book deals with orientable 3-manifolds. In particular, Lectures 11, 12, 13, 14, and 15 discuss the skein modules and algebras of orientable 3-manifolds. Lecture 16 discusses an invariant of these manifolds, known as the Witten-Reshetikhin-Turaev 3-manifold invariant. Example A.2.24 Trivial I-bundles over all unorientable surfaces, the twisted product ˆ 2 , and RP2 × S 1 are some examples of non-orientable of S 1 with S 2 , denoted by S 1 ×S 3-manifolds.

A.3 Smooth and PL Manifolds Topological manifolds can be equipped with additional structure, for example, a differentiable structure, which allows one to perform calculus on manifolds. This gives rise to an important class of manifolds known as smooth or differentiable manifolds.

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Definition A.3.1 (Smooth Manifolds) A function . f : Rn −→ Rm is said to be a .C q -function if it has continuous partial derivatives of all orders up to and including .q ≥ 0. The map f is said to be smooth or .C ∞ if it has continuous partial derivatives of all orders. A topological n-manifold is called a .C q -manifold if for any two charts .(Uα, φα ) and .(Uβ, φβ ) of M, the transition map .τα,β is a .C q -map. The manifold M is called a smooth or differentiable manifold if every transition map is smooth. In this case, the atlas . {(Uα, φα )} is said to be smooth, and the collection of all smoothly equivalent smooth atlases constitutes the smooth structure on M. By the definition above, .C 0 manifolds are the same as topological manifolds. All the topological manifolds described in Example A.2.6 can be equipped with a smooth structure. For example, the atlas, described in Example A.2.6.2, describes the standard smooth structure on .S n . Furthermore, the Cartesian product of smooth manifolds is smooth. Thus, .T n is a smooth n-manifold. We note that not all manifolds can be equipped with a smooth structure. Moreover, the smooth structure may not be unique. See Sect. A.3.1 for a discussion. An equivalent smooth definition of the orientability, which agrees with Definition A.2.19 for the orientability of topological manifolds, is as follows. Definition A.3.2 A smooth manifold M is said to be orientable if it has an atlas for which the Jacobian of all of its transition functions at any point on M has a positive determinant. In this case, the orientation of M is determined by this atlas. Otherwise, M is said to be non-orientable. If .(M, ∂ M) is oriented, then there is a natural orientation on .∂ M known as the induced orientation on .∂ M. If M is an oriented manifold, we can define another oriented manifold whose orientation is the reverse of the orientation of M. In particular, let .{(Uα, φα )}α∈Λ be the atlas that determines the orientation on M, where .φα : Uα −→ Rn such that . φα (x) = (x1, x2, . . . , xn ). Then, the atlas . {(Uβ , ψβ )}β ∈Λ of M, given by .ψβ : Uβ −→ Rn such that .ψβ (x) = (−x1, x2, . . . , xn ), determines the opposite orientation on M. We denote this manifold by .−M. Definition A.3.3 Let M and N be two smooth manifolds. A function . f : M −→ N is said to be a diffeomorphism if f is a homeomorphism and f and . f −1 are both smooth. Any two smooth manifolds are equivalent if they are diffeomorphic. Definition A.3.4 Let M and N be smooth manifolds with atlases .{(Uα, φα )} and {(Vβ, ψβ )}, respectively. A diffeomorphism . h : M −→ N is said to be orientation preserving if the Jacobian of the map .ψβ ◦ h ◦ φ−1 α has a positive determinant for all .α and . β at every point. If the Jacobian has a negative determinant for all .α and . β, then h is said to be orientation reversing. .

Manifolds are often studied from a category theoretic viewpoint, and it is essential to mention which category one is working in. The topological category, TOP, is the

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category whose objects are topological manifolds and morphisms are continuous functions. Manifolds can also be studied in other categories, for example, the smooth or differentiable category, DIFF, whose objects are differentiable manifolds and morphisms are smooth maps, and the piecewise linear category, PL, where the objects are PL manifolds and morphisms are PL maps. Most of the 3-manifold topology and knot theory described in this book takes place in the PL category which we now describe. Definition A.3.5 The standard n-simplex is a subset of .Rn+1 defined by .Δn = n  {(x0, x1, . . . , xn ) ∈ Rn+1 : xi = 1, xi ≥ 0 ∀i}. A face of .Δn is the convex hull of i=0

any non-empty subset of the .n + 1 coordinates that define the n-simplex. A zerodimensional face of .Δn is called a vertex of the simplex, while a one-dimensional face of .Δn is called an edge. Thus, .Δ0 is a point in .R; .Δ1 is a line segment connecting the points .(1, 0) and .(0, 1) in .R2 ; .Δ2 is an equilateral triangle in .R3 with vertices .(1, 0, 0), (0, 1, 0), and .(0, 0, 1); and .Δ3 is a tetrahedron in .R4 . See Fig. A.3 for an illustration.

Δ0

Δ1

Δ2

Fig. A.3: Illustration of standard simplices

Definition A.3.6 A simplicial complex .K is a topological space consisting of simplices that satisfy the following conditions: 1. Every face of a simplex in .K is also in .K. 2. If .si and .s j are two simplices in .K such that .si ∩ s j  , then .si ∩ s j is a face of both .si and .s j . The dimension of .K is the supremum of the dimensions of the simplices in .K. For a simplex s in .K, the star .st(s, K) of s with respect to .K is a subcomplex of .K, which consists of all simplices of .K which meet s, together with all their faces. The link .l k(s, K) of s is a subcomplex of .K consisting of all simplices that do not meet s but are faces of some simplex of .K containing s.3 3 This notation should not be confused with that for the linking number.

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 Definition A.3.7 Let . |K | = {s : s ∈ K}. A simplicial complex .K  is a subdivision of .K if . |K  | = |K | and each simplex of .K  lies in .K. For any two simplicial complexes .K and .L, a map . p : K −→ L is called a piecewise linear or PL map if there exist subdivisions .K  and .L  of .K and .L, respectively, such that under p, the vertices of .K  are sent to the vertices of .L  and the simplices of .K  are mapped linearly to the simplices of .L . Note that p is a PL homeomorphism if and only if .K and .L have a common subdivision. Definition A.3.8 Let .K be a simplicial complex. A triangulation of a topological manifold M is the pair .(K, h) such that . h : |K | −→ M is a homeomorphism. Thus, a triangulation of a topological manifold M is its subdivision into several simplices. Any two triangulations .(K1, h1 ) and .(K2, h2 ) of M are equivalent provided −1 .h 2 h1 : K1 −→ K2 is piecewise linear. A topological manifold equipped with a triangulation is called a triangulated manifold. Definition A.3.9 (Combinatorial Manifolds) A combinatorial or PL manifold is a triangulated manifold, possibly with boundary, such that the link of every simplex s of dimension p is PL homeomorphic either to the boundary of an .(n − p)-simplex or to an .(n − p − 1)-simplex. In this case, for any two charts .(Uα, φα ) and .(Uβ, φβ ) of M, the transition map .τα,β is a PL map. The collection of all PL equivalent PL atlases constitutes the PL structure on M. There is a difference between PL and triangulated manifolds. All PL manifolds can be triangulated; however, not all triangulated manifolds can be equipped with a PL structure. See Sect. A.3.1 for a discussion. The following definition of orientability for PL manifolds is equivalent to Definition A.3.2. Definition A.3.10 A PL n-manifold is said to be orientable if there is a choice of orientation for each n-simplex in the triangulation of M such that if s is an .(n − 1)-simplex adjacent to two n-simplices, then the orientations it inherits from them disagree. Thus, an oriented PL manifold is a PL manifold with a choice of orientation. If no triangulation of M admits such an orientation, then M is said to be non-orientable. Remark A.3.11 The orientability of a PL manifold is independent of the choice of its triangulation. See, for example, [RoSa, Hud].

A.3.1 Comparison of the TOP, DIFF, and PL Categories The question of potential relations between smooth, PL, and topological manifolds was posed by James W. Alexander in 1932 [Ale5]. As mentioned in [Kos], Alexander expressed confidence that the triangulation of smooth manifolds was only a question

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of “honest toil,” and later he asked about the validity of the so-called Hauptvermutung for manifolds and whether topological manifolds could be triangulated. In 1935, Pavel Sergeevich Alexandroff and Heinz Hopf [AH] asked whether every triangulated manifold carries a smooth structure. The Hauptvermutung, short for die Hauptvermutung der kombinatorischen Topologie, is German for “the main conjecture of combinatorial topology”.4 It states that the combinatorial topology of a simplicial complex .K is determined by the topology of the polyhedron . |K |. The manifold version of this conjecture states that any two triangulations of a PL manifold are combinatorially equivalent, that is, they become isomorphic after subdivision. This conjecture was first stated by Ernst Steinitz and Heinrich Tietze in 1908 in [Ste] and [Tie1], respectively. The polyhedral version of Hauptvermutung was verified for dimension 2 by C. D. Papakyriakopoulos in [Pap1]. The manifold version of the conjecture for dimension 2 was proved by Tibor Radó in 1925, while the 3-manifold version was proved by Edwin E. Moise in 1952 (see Theorems A.3.12 and A.3.13). Theorem A.3.12 ([Rad, Ker]) Every surface admits a PL structure, which is unique up to PL homeomorphism. Theorem A.3.13 ([Moi, Bin2, Ham]) Every topological 3-manifold admits a PL structure, which is unique up to PL homeomorphism. Moreover, every smooth manifold admits a canonical PL structure, which agrees with the smooth structure. The converse is true for 3-manifolds; however, it is not true in general (see Theorems A.3.14 and A.3.15 and [Kerv]). Theorem A.3.14 ([Cair1, Whi2]) Every smooth manifold admits a unique and compatible PL structure. If two smooth manifolds are diffeomorphic, then their triangulations are combinatorially equivalent. Theorem A.3.15 ([Whi4]) Every PL 3-manifold has a smooth structure, which is unique up to diffeomorphism. Together with Theorem A.3.13, this implies that every topological 3-manifold admits a unique smooth structure. Theorems A.3.12–A.3.15 imply that the concepts of topological, smooth, and PL 3-manifolds coincide. Thus, any topological manifold whose dimension is less or equal to 3 has a smooth and a PL structure. These structures are unique in that any two smooth or PL manifolds that are homeomorphic to M are also diffeomorphic or PL homeomorphic to each other, respectively. 4 At the International Congress of Mathematicians held in Zürich in 1932, Alexander stated that it as one of the major problems in topology.

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When .n ≥ 4, however, the notions of TOP, DIFF, and PL start to diverge. Michel A. Kervaire was the first to show this, when in 1960 he gave an example of a tendimensional combinatorial manifold that admits no differentiable structure [Kerv]. In 1956, John Willard Milnor exhibited the existence of smooth homotopy 7-spheres5 that are homeomorphic to .S 7 but not diffeomorphic to it [Mil1]. Such manifolds are called exotic spheres. In 1963, Kervaire and Milnor showed that .S 7 admits exactly 28 non-diffeomorphic structures [KM]. Milnor’s discovery of exotic spheres in dimension 7 disproved the smooth Poincaré conjecture in dimension 7. The Poincaré conjecture was posited by Poincaré [Poi2] in 1904, when he discovered a 3-manifold whose homology groups were the same as those of .S 3 , but whose fundamental group was nontrivial.6 He asked whether a 3-manifold with the same homology groups and fundamental group as .S 3 was indeed .S 3 . Conjecture A.3.16 (The Poincaré Conjecture/Perelman’s Theorem) Any closed 3-manifold that is homotopy equivalent to .S 3 is also homeomorphic to it. This conjecture remained unsolved for almost a century. Many mathematicians attempted to prove this conjecture in the twentieth century, the most famous attempt of which we feel was by Whitehead in the 1930s. He was quick to retract his proof, and in the process, he discovered the existence of an open contractible 3-manifold that is not homeomorphic to .R3 [Whi1]. This manifold is now known as the Whitehead manifold.7 The Whitehead manifold is constructed as follows. Let T be the standard solid torus in .S 3 , and consider the homeomorphism . h : S 3 −→ S 3 which takes T onto the solid torus . h(T) and where . h(T) lies inside of T as illustrated in Fig. A.4. We can continue in this fashion to obtain an infinite nested sequence of solid tori ∞  2 3 .T , h(T), h (T), h (T), . . .. Let . X = hi (T). Then, the Whitehead manifold is defined as .W = S 3  X.

i=0

The Poincaré conjecture was ultimately proved by Grigori Perelman in the year 2003 [Per1, Per2, Per3]. One can formulate Poincaré’s conjecture for dimensions other than 3. The classification of surfaces (see Theorem A.5.11) shows that the Poincaré conjecture is true for dimension .n = 2. In fact, the Poincaré conjecture can be generalized and formulated in any dimension and in the PL and DIFF categories as well.

5 A homotopy n-sphere is a closed n-manifold that is homotopy equivalent to .S n . 6 This manifold is now known as the Poincaré homology sphere. Its fundamental group is the binary icosahedral group . 2, 3, 5 whose order is 120. This manifold is also discussed in Sect. A.7. 7 The Kauffman bracket skein module of the Whitehead manifold is discussed in Lecture 12.

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h( )

Fig. A.4: The solid torus T with . h(T) lying inside it

Conjecture A.3.17 (The Generalized Poincaré Conjecture) Every topological (respectively, smooth or PL) homotopy n-sphere is homeomorphic (respectively, diffeomorphic or PL homeomorphic) to .S n . This conjecture is true in the TOP category in all dimensions. Stephen Smale proved the conjecture in the affirmative for dimensions .n ≥ 5, while Michael E. Freedman proved the conjecture for dimension .n = 4 (see [Sma2, Sma3], and [Fre]). The book [BKKPR] contains a thorough exposition of the disk embedding theorem for four-dimensional manifolds, which is foundational to virtually all our understanding of topological 4-manifolds, and, most famously, includes the fourdimensional Poincaré conjecture in the TOP category. Stewart Scott Cairns [Cair2] and Morris W. Hirsch [Hir] showed that every PL 4-manifold has a unique smooth structure and vice versa. Thus, in four dimensions, the PL and DIFF categories are the same, which implies that the PL and DIFF versions of the generalized Poincaré conjecture are equivalent. The smooth four-dimensional Poincaré conjecture has not been proved at the time of writing this book (however, see [Akb]). In fact, the PL version of Conjecture A.3.17 is true for all dimensions except 4. For .n ≥ 5, this was proved by Smale. The smooth version of Conjecture A.3.17 is known to be true only in a handful of dimensions and is false in general. In the 1980s, Freedman and Simon Donaldson showed the existence of exotic R4 , that is, they showed that .R4 has more than one smooth structure (see [Fre] and [Don]. In 1987, Clifford Henry Taubes showed that .R4 admits a continuum number of nonequivalent smooth structure [Tau]. However, not all topological manifolds admit a smooth or PL structure. A four-dimensional example was given by Freedman and is known as Freedman’s example or the .E8 manifold8 [Fre]. This manifold does not admit any PL structure and is not even triangulable as a simplicial complex. In [KiSi], Robion C. Kirby and Laurence C. Siebenmann exhibited the existence of manifolds without piecewise linear structures in any dimension greater than 4. They also answered in the negative the Hauptvermutung for manifolds about the uniqueness for PL structures.9 Furthermore, for dimension 4, the notions of

.

8 This manifold is known as .E8 , since its intersection form is .E8 with signature 8. 9 The manifolds .S 3 × R2 and .S 3 × S 1 × S 1 admit nonequivalent PL structures.

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triangulated and combinatorial manifolds coincide. However, the manifold .E8 × S 1 is a five-dimensional manifold that does not admit a PL structure but is triangulable as a simplicial complex (see, e.g., [Rud]). In 2013, Ciprian Manolescu showed the existence of compact topological n-manifolds, for .n ≥ 5, that admit no triangulation [Man]. In summary, TOP, DIFF, and PL are equivalent for dimensions .n ≤ 3. For .n = 4, PL and DIFF are equivalent, while TOP differs. For .n ≥ 5, they all start to differ. We end this subsection with the following theorem, which is true for smooth and PL manifolds but not for topological manifolds, even in dimension 3. Theorem A.3.18, along with Theorems A.3.13 and A.3.15, allows us to move freely between DIFF and PL and between isotopy and ambient isotopy in dimension 3. Theorem A.3.18 ([Hud, Kos]) Let M and N be two differentiable or PL manifolds of any dimension, and let M be compact. Then any two embeddings .φ, ψ : M → N are isotopic if and only if they are ambient isotopic. There are several ways in which one can decompose 3-manifolds into simpler pieces so as to ease the study of 3-manifolds. In this appendix, we briefly discuss the handle decomposition, Heegaard decomposition, prime decomposition, and JSJ decomposition of 3-manifolds.

A.4 Handle Decomposition and Heegaard Decomposition Smale’s theory of handles and handlebodies has provided topologists with a very successful method for decomposing smooth and, in particular, three-dimensional manifolds. He showed that every closed manifold can be constructed by successive attachment of handles. This enabled him to prove the h-cobordism theorem and the generalized Poincaré conjecture in dimensions 5 and higher (see [Sma2] and [Sma3]). His work is a generalization of that of August Ferdinand Möbius [Mob2], who used Morse functions in a similar way as Smale to decompose surfaces. We briefly describe handles and handle decomposition. Definition A.4.1 (Handles and Handle Addition) Let W be an .(n + 1)-dimensional manifold and M an n-dimensional manifold embedded in .∂W. We define a p-handle addition to W along M as follows: Consider the .(n + 1)-disk . D n+1 = D p × D n+1−p . This product structure of . D n+1 is called a p-handle. Its boundary is .∂D n+1 = (∂D p × D n+1−p ) ∪ (D p × ∂D n+1−p ), and since .S p−1 = ∂D p , .∂D n+1 = (S p−1 × D n+1−p ) ∪ (D p × S n−p ). Consider an embedding . f : S p−1 × D n+1−p −→ M. We form a new manifold .W  by gluing . D n+1 to W using the map f , and we say that .W  is obtained from W by p-handle addition

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to W along M. . D p × {0} is called the core of the handle, while .{0} × D n+1−p is called the cocore of the handle. When .n + 1 = 3, a 0-handle addition consists of adding a disjoint 3-ball to a 3-manifold W, since .∂D0 × D3 = ∅. A 1-handle addition is the gluing of the solid cylinder . D1 × D2 to W along .S 0 × D2 , that is, along its two boundary disks, while a 2-handle addition is attaching a “plate” . D2 × D1 to W along the annular boundary 1 1 3 0 2 0 . S × D . Finally, a handle of index 3, . D × D , is a 3-ball glued to W along . S × D , often visualized as capping off an entire hole. See Fig. A.5 for an illustration. 2-handle 1-handle 0-handle

3-handle

Fig. A.5: Handle additions when .n + 1 = 3 and . p = 0, 1, 2, and 3

Remark A.4.2 A surgery of index p on the n-manifold M is defined as a modification of M when a p-handle is added along .∂W = M. For example, if M is a 3-manifold, a surgery of index 2 on M is obtained by drilling out from M the solid torus 1 2 2 2 2 1 . S × D = ∂D × D and replacing this with another solid torus . D × S . The new meridian of the solid torus intersects the old meridian at one point. For .n = 3, this is a special integral case of Dehn surgery, which is discussed in Appendix B. Definition A.4.3 A handle decomposition of a compact n-manifold is its decomposition into p-handles, . p ≥ 0, the union of which is M.10 Using Morse theory, the following result was proved about handle decomposition. Theorem A.4.4 ([Sma4]) Every closed, oriented, smooth n-manifold admits a handle decomposition. 10 Handle decomposition for manifolds is analogous to CW decomposition for CW complexes.

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For the PL case, handle decomposition follows easily from the triangulation of a manifold (see, e.g., [RoSa]). Example A.4.5 .S 3 is obtained from a 0-handle and a 3-handle by gluing the 3-handle along the boundary of the 0-handle Example A.4.6 The handle decomposition of .T 3 consists of one 0-handle, three 1handles, three 2-handles, and one 3-handle. While building manifolds, we glue handles of lower indices first, and handles of the same index are added simultaneously. We can also reverse this procedure to obtain a dual handle decomposition for which a handle of index p becomes a handle of index .n − p. The core of a p-handle is the cocore of the corresponding dual .n − p handle. Definition A.4.7 A handlebody .Hn is a connected, compact, orientable 3-manifold with boundary obtained by attaching n copies of the 1-handle . D1 × D2 to the 3-ball 3 2 1 . D . The gluing homeomorphisms match the 2n disks . D × ∂D with 2n disjoint 3 2 2-disks in .∂D = S so that the resulting manifold is orientable. The positive integer n is called the genus of the handlebody .Hn . The boundary of .Hn is homeomorphic to an oriented surface of genus n. Note that .Hn  F0,n+1 × [0, 1], where .F0,n+1 denotes a 2-sphere with .n + 1 boundary components. Notice that the handle decomposition of any handlebody of genus g consists only of one 0-handle and g 1-handles. The idea of decomposing a closed oriented 3-manifold along the boundaries of two handlebodies was first thought of by Poul Heegaard in his PhD thesis [Hee]. Definition A.4.8 A Heegaard splitting of genus n of a closed oriented 3-manifold M is its decomposition into two handlebodies .Hn and .Hn such that . M = Hn ∪F Hn , where .F = Hn ∩ Hn = ∂Hn = ∂Hn . Here, M is obtained by a gluing .Hn to .Hn via an orientation-reversing homeomorphism . f : ∂Hn −→ ∂Hn . The surface F is known as the Heegaard surface or splitting surface of M. The genus of a Heegaard splitting is the genus of F.11 Remark A.4.9 The definition of Heegaard splittings can be extended to nonorientable 3-manifolds where we take non-orientable handlebodies and glue them along their common non-orientable boundary surfaces. The solid Klein bottle is an example of a non-orientable handlebody. Any two Heegaard splittings of M are said to be equivalent or isotopic if their splitting surfaces are ambient isotopic. 11 The structure of the Kauffman bracket skein module of a 3-manifold based on its handle and Heegaard decompositions is discussed in Lecture 12.

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Definition A.4.10 The minimal genus over all Heegaard splittings of M is called the Heegaard genus of M. If M has Heegaard genus n, then it admits Heegaard splittings of all genera .m > n. From Theorem A.3.13, we obtain the following result. Theorem A.4.11 ([Moi]) Every closed orientable 3-manifold admits a Heegaard splitting. Example A.4.12 .S 3 is an example of a 3-manifold with Heegaard genus 0. This is because .S 3 can be obtained by gluing two genus zero handlebodies (3-balls) along the Heegaard surface, which is a 2-sphere. In fact, the only 3-manifold whose Heegaard genus is 0 is .S 3 . Compare with Alexander’s trick discussed in Theorem B.1.2. In 1968, Friedhelm Waldhausen showed that, up to isotopy, .S 3 admits a unique Heegaard decomposition of every genus (see Theorem A.4.16 for its generalization to lens spaces). In particular, .S 3 admits a genus 1 Heegaard splitting: gluing two solid tori .H1 and .H1 along their boundaries such that the meridian of .H1 is identified with the longitude of .H1 results in .S 3 . Example A.4.13 Consider two solid tori .H1 and .H1 glued along their boundaries by an orientation-reversing homeomorphism that identifies the meridian of .∂H1 with a  .(p, q) torus knot in . ∂H , where p and q are relatively prime. This gluing describes 1 the Heegaard genus 1 splitting of an important class of closed oriented 3-manifolds called lens spaces, denoted by . L(p, q). The handle decomposition of lens spaces is as follows: start with a 0-handle, and glue a 1-handle to it to get a solid torus. Then attach a 2-handle to obtain a lens space with a hole, and finally, add a 3-handle to cap off this hole. The 3-manifold .S 1 × S 2 is obtained by gluing two solid tori by the identity homeomorphism on their boundaries, and it is homeomorphic to . L(0, 1). However, it is often considered a degenerate case of lens spaces as is .S 3 , which is the homeomorphic to . L(1, 0). .S 1 × S 2 and lens spaces are the only 3-manifolds whose Heegaard genus is 1. Exercise A.4.14 1. Show that an .S 1 bundle over .S 2 has a Heegaard decomposition of genus 1. 2. Which lens space is obtained as the tangent circle bundle over .S 2 ? Hint to (1): .S 2 has an atlas composed of two 2-disks. See Exercise A.2.7. Answer to (2): We get . L(2, 1). Example A.4.15 The Heegaard genus of Seifert fibered manifolds is at least two. These manifolds will be discussed briefly in Sect. A.7.

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Theorem A.4.16 ([BO]) .

Every lens space . L(p, q) has a unique Heegaard decomposition of every genus n ≥ 1.

Definition A.4.17 A stabilization of a genus n Heegaard splitting of a 3-manifold M = Hn ∪F Hn is a genus .n + 1 Heegaard splitting of M obtained by adding an unknotted12 1-handle to the Heegaard surface F. This transforms the genus n handlebodies .Hn and .Hn into handlebodies of genus .n + 1 and thus gives a new splitting of the manifold M. Any two Heegaard splittings of M are said to be stably equivalent if they become equivalent after a finite number of stabilization operations on each of them.

.

Example A.4.18 The genus 1 splitting of .S 3 described in Example A.4.12 can be obtained from its genus 0 splitting by a stabilization. Remark A.4.19 Stabilizations can be understood using the language of connected sums. Let .Fi be the Heegaard surface of . Mi , .i = 1, 2, respectively. We define the connected sum .(M, F) = (M1, F1 ) # (M2, F2 ) as follows. Choose a point . xi ∈ Fi and small neighborhoods . Bi3 of . xi . We remove the interiors of . Bi3 from .(Mi, Fi ) and glue the resulting manifolds along .∂Bi3 by an orientation-reversing homeomorphism. If .(M2, F2 ) is the standard genus 1 Heegaard decomposition of .S 3 , then .(M, F) is a stabilization of .(M2, F2 ). See Sect. A.5 for a discussion of connected sums of manifolds. Theorem A.4.20 ([Rei3, Sin]) Any two Heegaard splittings of a closed oriented 3-manifold are stably equivalent.

A.4.1 Alternative Definitions and the Classification of Lens Spaces There are several equivalent definitions of lens spaces, of which we describe two. Definition A.4.21 Let .S 3 = {(z1, z2 ) : |z1 | 2 + |z2 | 2 = 1} be considered as the unit sphere in .C2 (or .R4 ), and consider the free .Z p action on this .S 3 given by the self-homeomorphism .τ on .S 3 under which .(z1, z2 ) −→ (e2πi/p z1, e2πiq/p z2 ). The orbit space of this action is the lens space . L(p, q). That is, . L(p, q) = S 3 /∼, where k 3 .u ∼ v ⇐⇒ v = τ (u) for some k, and .u, v ∈ S .

12 This is a handle whose core is parallel to the Heegaard surface. More precisely, a 1-handle is unknotted if there is a disk .D 2 in M such that .D ∩ Hn+1 = ∂D 2 and curve .∂D 2 goes along the 1-handle only once.

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Definition A.4.22 Consider the unit 3-ball . D3 in .R3 . Identify each point on the upper half of .∂D3 with its image under counterclockwise rotation by an angle of .2πq/p about the z-axis followed by a reflection about the xy-plane. See Fig. A.6 for an illustration. The 3-manifold thus obtained is the lens space . L(p, q).

2

Fig. A.6: . L(p, q) as the identification space of . D3

Remark A.4.23 The image of a point in the upper half of .∂D3 under counterclockwise rotation by an angle of .π about the z-axis, followed by a reflection about the xyplane, is its antipodal point. This identification occurs when . p = 2 and .q = 1. Thus, 3 .RP  L(2, 1) (compare with Example A.2.6.4). The earliest considerations of lens spaces different from .RP3 appear in the work by Heegaard (see Lecture 2), and their systematic study and formal name were given by Tietze in 1908. They constitute an important class of closed 3-manifolds that have been completely classified. They are also the only known examples of 3-manifolds that are not completely determined by their homotopy type. In 1919, Alexander showed that . L(5, 1) and . L(5, 2) are not homeomorphic to each other even though they have the same fundamental group .Z5 . Reidemeister was the first to classify lens spaces up to PL homeomorphisms using what is now known as “Reidemeister torsion”. Theorem A.4.24 ([Rei4]) Two lens spaces . L(p, q) and . L(p, q ) are homeomorphic if and only if . p = p and ≡ ±q±1 mod p. In addition, the lens space . L(p, q) allows an orientation-reversing self-homeomorphism if and only if . p = 0, 1, 2 or .q2 ≡ −1 mod p. In particular, the lens space . L(3, 1) does not allow an orientation-reversing homeomorphism.

 .q

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In [Bro], E. J. Brody proved the same result for lens spaces in the TOP category. Theorem A.4.25 ([Whi3]) Two lens spaces . L(p, q) and . L(p, q ) are homotopy equivalent if and only if    2 . p = p and .±qq is a quadratic residue modulo p, that is, .±qq ≡ ±m mod p, for some integer m. Thus, for example, the lens spaces . L(7, 1) and . L(7, 2) are homotopy equivalent but not homeomorphic to each other. In 2003, Akira Yasuhara and the first author gave a purely knot theoretic classification of lens spaces (see [PY] for more details). The lens space . L(p, q) can also be constructed by Dehn surgery along links in .S 3 , the simplest of which is the unknot with surgery coefficient . −p q . This construction will be discussed in detail in Appendix B.

A.5 Prime Decomposition As discussed earlier, there are several ways of classifying and studying 3-manifolds, most of which involve breaking 3-manifolds into “smaller” pieces. In Sect. A.4, we studied 3-manifolds by decomposing them into handles and handlebodies. We now discuss prime decomposition in which 3-manifolds are decomposed along 2-spheres. Definition A.5.1 (Connected Sums) The connected sum . M1 # M2 of two connected n-manifolds . M1 and . M2 is constructed by identifying . M1 \int(D n ) and . M2 \int(D n ) along the common boundary .(n−1)-spheres. If both . M1 and . M2 are oriented, then the identification of the boundary spheres is given by an orientation-reversing homeomorphism. This construction is uniquely defined. We stress that the connected sum operation depends on the choice of orientation of . M1 and . M2 . There do exist orientable manifolds . M1 and . M2 such that . M1 # M2  M1 # (−M2 ), where .−M2 is obtained from . M2 by reversing its orientation. We can find examples of such manifolds in dimension 3. By the uniqueness of prime decomposition of oriented 3-manifolds (see Theorem A.5.10), it is enough to consider orientable 3-manifolds, neither of which admits an orientation-reversing self-homeomorphism. Lens spaces . L(p, q), for which .q2 ≡ −1 mod p, are such a class of 3-manifolds (see Theorem A.4.24). Figure A.7 illustrates the connected sum of two tori. Exercise A.5.2 Show that .Hn # Hm is obtained by gluing a 2-handle to .Hn+m along the annular neighborhood of the boundary of the disk that separates the n toroidal holes from the remaining m toroidal holes. See Fig. A.8 for the case .n, m = 1.

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Fig. A.7: The interior of . D2 is removed from each .T 2 , and the resulting manifolds are glued along their common circle boundaries to obtain .T 2 # T 2

This exercise has ramifications for the computation of Kauffman bracket skein modules of connected sums of 3-manifolds. See [BLP, BaPr], and Example 12.4.11 in Lecture 12.

Fig. A.8: .H1 # H1 is obtained by attaching a 2-handle to .∂H2 along the curve .γ. Note that . D ∪γ Dγ gives rise to the separating sphere of the connected sum .H1 # H1

Exercise A.5.3 Show that .S 3 is the identity element of the connected sum operation for 3-manifolds. Definition A.5.4 (Prime Manifolds) A connected 3-manifold M is prime if . M = M1 # M2 implies either . M1 = S 3 or 3 . M2 = S . Lens spaces, .R3 , handlebodies, and .T 3 are some examples of prime 3-manifolds. is also a prime manifold. This is due to a theorem proved by Alexander which states that every embedded 2-sphere in .S 3 bounds an embedded 3-ball. However, there are many mathematicians who do not view .S 3 as prime, much like the number

3 .S

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1. The following proposition about 3-manifolds whose boundary contains .S 2 has ramifications in Lecture 12, where the skein module of the connected sum of 3manifolds is computed. Proposition A.5.5 Let M be an oriented 3-manifold such that .∂ M = S 2 . Then 3 . M = M1 # D , for some oriented 3-manifold . M1 . Definition A.5.6 (Irreducible Manifolds) A 2-sphere in a 3-manifold M is said to be incompressible if it does not bound a 3-ball in M; otherwise, it is said to be compressible. Non-separating 2-spheres in M are incompressible 2-spheres. The manifold M is irreducible if every 2-sphere in M is compressible. If M contains an incompressible 2-sphere, then it is said to be reducible. The following proposition gives us a useful criterion for identifying irreducible manifolds.  −→ M be a covering space. Then M is irreducible if Proposition A.5.7 Let . p : M  is irreducible. The converse is true if M is orientable. M

.

Thus, manifolds whose universal covers are .R3 or .S 3 are irreducible. Exercise A.5.8 1. Show that .T 3 is irreducible. 2. Show that all hyperbolic 3-manifolds are irreducible. Hint for (1) and (2): Show that the universal cover of .T 3 and all hyperbolic manifolds is .R3 . The irreducibility of a manifold is a slightly stronger property than that of primeness. All irreducible 3-manifolds are prime. However, the converse is not true. We formalize these statements in the theorem below. Theorem A.5.9 With the exceptions of .S 1 × S 2 and .S 1 ׈ S 2 , any 3-manifold is prime if and only if it is irreducible. The manifolds .S 1 × S 2 and .S 1 ׈ S 2 are prime and reducible. We now discuss the existence and uniqueness of the decomposition of 3-manifolds into connected sums. Theorem A.5.10 Every compact orientable 3-manifold factors as a connected sum of a finite number of prime 3-manifolds, and this decomposition is unique up to homeomorphism and the ordering of prime factors.

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The existence of prime decomposition was proven by Hellmuth Kneser, but it was Wolfgang Haken who first formulated the exact result and proved the uniqueness of this decomposition. For a nice concise proof for closed oriented 3-manifolds, see [Mil3]. Milnor’s proof also works for oriented 3-manifolds with boundary (see [Hem]). Every compact non-orientable 3-manifold can also be written as a connected sum of a finite number of prime 3-manifolds; however, the existence and uniqueness of this decomposition are more subtle (see [JaPr]). This is because 1 2 1 ˆ S 2 ), for any non-orientable 3-manifold M. Notice . M # (S × S )  M # (S × that this is the three-dimensional analogue of the two-dimensional property that 2  F # K b for any non-orientable surface F. The proof for the structure .F # T theorem for the connected sum of compact 3-manifolds given in [Mil3] requires some modifications (see [JaPr]). Hence, the set of all compact, connected 3-manifolds with the connected sum operation forms an infinitely generated commutative monoid with .S 3 as the identity element. The generators of this monoid are prime 3-manifolds, and the relators are 1 2 1 ˆ S 2 ), for any non-orientable 3-manifold M. Therefore, . M # (S × S )  M # (S × compact connected oriented 3-manifolds form a free commutative monoid generated by oriented prime manifolds under the connected sum operation. A similar result holds for surfaces, which we now state. Theorem A.5.11 (Classification of Surfaces) The set of compact connected surfaces under the connected sum operation, .(F , #), forms a commutative monoid with three generators as follows. (F , #) = D2, T 2, RP2 | T 2 # RP2  RP2 # RP2 # RP2 .

.

S 2 is the identity element of this monoid. In particular, the submonoid of oriented surfaces is a free abelian monoid with two generators .T 2 and . D2 . .

The number of tori involved in the connected sum for an orientable surface is known as its genus. For an unorientable surface, the number of real projective planes in its connected sum presentation is its genus. The proof of Theorem A.5.11 has been known since the 1860s. Möbius was the first to provide an outline of the proof of this result in [Mob1]. The first rigorous proof of the classification theorem for compact surfaces is ascribed by some to Henry Brahana [Bra]. In [Mas], a careful proof of this result, following [Bra], is given using the triangulation of surfaces. However, the book [ST] attributes the proof to Dehn and Heegaard [DH] and does not list Brahana’s paper in its bibliography. We finish this section with a result by Haken about the Heegaard genus of the connected sums of 3-manifolds. Theorem A.5.12 ([Hak2]) Let . Mi be closed orientable 3-manifolds with Heegaard genera .gi , .1 ≤ i ≤ n. n  Then the Heegaard genus of the manifold . M = M1 # M2 # . . . # Mn is . gi . 1

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A.6 Haken Manifolds An important class of 3-manifolds, now known as Haken manifolds, was introduced by Haken in the year 1961 in [Hak1]. He pioneered the study of incompressible surfaces (see Definition A.6.4) in 3-manifolds. We introduce some preliminary definitions needed for defining Haken manifolds. Definition A.6.1 A submanifold .(N, ∂N) of .(M, ∂ M), allowing .∂N = ∅, is said to be properly embedded in .(M, ∂ M) if under the inclusion map .∂N is mapped to .∂ M and .int(N) is mapped to .int(M). Therefore, a surface F is properly embedded in a 3-manifold M if .F ∩ ∂ M = ∂F. Definition A.6.2 A properly embedded codimension 1 submanifold F of a manifold M is said to be two-sided if its normal bundle is trivial, that is, .∃ an embedding . h : F × [−1, 1] −→ M such that . h(x, 0) = x for all . x ∈ F and . h(F × [−1, 1]) ∩ ∂ M = h(∂F × [−1, 1]). We now define the notion of compressibility of surfaces, different than .S 2 and in 3-manifolds.

2 .D ,

Definition A.6.3 Let F be a properly embedded surface in a 3-manifold M or a part of .∂ M. A disk . D2 in M is called a compressing disk of F if . D2 ∩ F = ∂D2 and 2 2 . ∂D is not contractible in F, that is, . ∂D does not bound a another disk in F. Definition A.6.4 A properly embedded surface F in M that admits a compressing disk is said to be compressible. If the surface F is not the disjoint union of .S 2 ’s or . D2 ’s and it contains no compressing disk, then the surface is said to be incompressible. If .F = S 2 , then F is incompressible if it does not bound a 3-ball in M. If .F = D2 , then F is incompressible if it does not bound another disk in .∂ M. As a special case, if F is a properly embedded two-sided surface in M such that F is neither a sphere nor a disk, then F is incompressible if and only if the map .π1 (F) −→ π1 (M) induced by the inclusion is injective. Definition A.6.5 A properly embedded surface .F ⊂ M is said to be parallel to the boundary of M if their exists an embedding . h : F × [0, 1] −→ M such that . h(F × {0}) ⊂ M and . h(F × {1}) = F ⊂ M, in the case when . ∂F = ∅. If . ∂F  ∅, F × [0, 1] −→ M. then . h : (x, t) = (x, 0), for x ∈ ∂F Definition A.6.6 An irreducible manifold M is said to be atoroidal if every incompressible torus in M is parallel to the boundary of M. Definition A.6.7 A properly embedded surface F in a 3-manifold M is said to be essential if one of the following holds:

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1. F is a 2-sphere that does not bound a 3-ball. 2. F is a disk with .∂F ⊂ ∂ M not bounding a disk in .∂ M. 3. F is not a disk or sphere and is incompressible and not parallel to the boundary of M. Therefore, an irreducible 3-manifold contains no essential 2-spheres, and an atoroidal 3-manifold contains no essential tori. Definition A.6.8 A 3-manifold is .P2 -irreducible if it is irreducible and contains no two-sided projective planes. Definition A.6.9 A .P2 - irreducible 3-manifold M is sufficiently large if it contains a properly embedded incompressible surface F with .F  S 2, D2 . Definition A.6.10 A compact, sufficiently large 3-manifold is called a Haken manifold, that is, a Haken manifold is a compact, .P2 -irreducible three-dimensional manifold which contains a properly embedded, incompressible, two-sided surface. If one considers only orientable Haken manifolds, then .P2 irreducibility will simply be replaced by irreducibility. The complements of nontrivial knots in .S 3 , and .F ×S 1 , where .F  S 2 is a compact oriented surface, are all Haken manifolds. However, .S 3 , .S 1 × S 2 , lens spaces, and many Seifert fibered manifolds, which are defined in the following section, are not Haken manifolds. Haken manifolds are an important class of 3-manifolds that have been completely classified. In particular, using the Heegaard decomposition of 3manifolds, it has been shown that there is an algorithm that determines whether a 3-manifold is Haken or not (see [JO]).

A.7 Seifert Fibered Manifolds and JSJ Decomposition The class of circle bundles over surfaces can be extended by introducing singular fibers. These are known as Seifert fibered manifolds and were introduced by Herbert Seifert in the year 1933. Definition A.7.1 Consider an orientable surface of genus g with b boundary components, which we denote by .Fg,b . Let . x1, x2, . . . , xn ∈ Fg,b and . D12, D22, . . . , Dn2 denote their regular neighborhoods in the interior of .Fg,b , respectively. Consider the  n  int(Di2 ) × S 1 , which is obtained by drilling the solid manifold .(Fg,b × S 1 ) \ i=1

tori .i nt(Di2 ) × S 1 out of .(Fg,b × S 1 ). This is a compact orientable 3-manifold whose boundary consists of n tori .ai × S 1 , where .ai denote the curves .∂Di2 . Now consider n pairs of relatively prime integers .(p1, q1 ), (p2, q2 ), . . . , (pn, qn ) with . pi ≥ 1. Glue

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the n solid tori back to the 3-manifold so that the meridian . μi of the .i th solid torus is glued to a curve on .∂Di2 × S 1 such that . μi = pi · ai + qi · t in .H1 (∂Di2 × S 1 ). Here, t represents the regular fiber. For each i, the image under the gluing of the curve 1 . xi × S is called the ith singular fiber if . pi > 0. . pi is the multiplicity of the singular fiber . xi × S 1 . The manifold obtained by this construction is called a Seifert fibered manifold fibers and oriented base .Fg,b . We denote this manifold by with nq1 singular q2 qn

, , . . . , . The qualifier O in the notation signals the fact that the . M O, g, b; p1 p2 pn base surface is orientable. Seifert fibered manifolds with one or two singular fibers have Heegaard genus 1.

In fact, . M O, 0, 0; qp11 is the lens space . L(q1, p1 ). Seifert fibered manifolds with two special fibers are also lens spaces. A Seifert fibered manifold with three special fibers has Heegaard genus 2. A quintessential example of such a Seifert fibered manifold

1 1 with oriented base .S 2 is the Poincaré homology sphere, . M O, 0, 0; −1 2 , 3 , 5 . This  manifold is often denoted by . (2, 3, 5). Moreover, the special class of manifolds, which include certain prism manifolds and the quaternionic manifold, discussed in In fact, prism Example 12.4.10 in Lecture 12 are also Seifert fibered manifolds.

manifolds are the Seifert fibered manifolds . M O, 0, 0; 12 , 12 , qp . Seifert fibered manifolds with oriented base .S 2 and three special fibers are called a small Seifert fibered manifold. Such manifolds are Haken manifolds if and only if their Euler number .e = qp11 + qp22 + qp33 is zero. Beyond prime decomposition, there is a further canonical decomposition, known as the canonical torus decomposition or JSJ decomposition, of irreducible compact orientable 3-manifolds. In this decomposition, we split 3-manifolds along tori rather than spheres. The idea of JSJ decomposition for 3-manifolds began with the work of Waldhausen in [Wal2] and was developed later through the work of William H. Jaco, Peter B. Shalen, and Klaus Johannson in the 1970s. The acronym JSJ refers to Jaco, Shalen, and Johannson. Theorem A.7.2 (JSJ Decomposition [JS, Joh]) If M is an irreducible compact orientable manifold, then there is a collection of disjoint incompressible tori .T1, T2, . . . , Tn in M such that splitting M along the union of these tori produces manifolds . Mi which are either Seifert fibered or atoroidal— every incompressible torus in . Mi is isotopic to a torus component of .∂ Mi . Furthermore, a minimal such collection of tori .Tj is unique up to isotopy in M. From the work of Thurston and Perelman, it follows that the atoroidal pieces are either Seifert fibered or hyperbolic. See [Ada, Mart], and [Thur2] for more details about hyperbolic geometry.

Appendix B

Surgery on Links in the 3-Sphere and Kirby Calculus

In this appendix, we familiarize the reader with the mapping class group of surfaces. We define Dehn twist homeomorphisms and outline various examples for the mapping class group of orientable and non-orientable surfaces. The appendix then gives a detailed description of surgery on links and the Kirby calculus, discusses the linking matrix, and finishes with a brief description of parallelizable 3-manifolds.

B.1 The Mapping Class Group of Surfaces Definition B.1.1 ([Lic8]) Let X be a finite simplicial complex, G X be the group of piecewise linear homeomorphisms mapping X to X, and NX be the subset of G X such that all elements of NX are isotopic to the identity. Then NX  G X , and G X /NX is the mapping class group, denoted by Mod± (X). In the examples to follow when we work with orientable surfaces, Fg,d , we will consider the group of piecewise linear orientation-preserving homeomorphisms. In this case, we will denote the mapping class group by Mod+ (Fg,d ), and we will call Mod± (Fg,d ) the extended mapping class group. For the mapping class group of unorientable surfaces F, we will use the notation Mod(F). Theorem B.1.2 (Alexander’s Trick [Ale3, Smit, Tie2]) Mod+ (D2 ) = {1}. Proof Let D2 be defined in C as the unit disk {z ∈ C; |z| ≤ 1}, and let h : D2 → D2 be a piecewise linear orientation-preserving homeomorphism, and suppose h is fixed on ∂D2 , that is, h(z) = z when |z| = 1. We will define an isotopy H(z, t) : D2 ×[0, 1] → D2 between h and id D 2 by using Alexander’s trick where H(z, 1) = h(z) and H(z, 0) = z. That is,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5

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(1 − t)h . H(z, t) = z



z

1−t

for 0 ≤ |z| < 1 − t, for 1 − t ≤ |z| ≤ 1. 

Remark B.1.3 While the generalization of James W. Alexander’s theorem to all n for the piecewise linear case is immediate, it is not the case for smooth diffeomorphisms. Stephen Smale in [Sma1] proved Mod+ (D2 ) = {1} for smooth orientationpreserving diffeomorphisms, and he conjectured that the group of diffeomorphisms of the 3-sphere is homotopy equivalent to the orthogonal group O(4). In 1983, Allen E. Hatcher in [Hat1] proved Smale’s conjecture for D3 which implies that Mod+ (D3 ) = {1}. Smale’s conjecture for D n in higher dimensions is the claim that the group of self-diffeomorphisms of D n is contractible. In 2018, Tadayuki Watanabe in [Wat] released a preprint that proves the failure of Smale’s conjecture in the four-dimensional case. For n ≥ 5, it is well known, mainly by the work of Michel A. Kervaire and John W. Milnor in [KM], that Smale’s conjecture fails. Theorem B.1.4 ([Sma1]) The space Ω of all orientation-preserving C ∞ diffeomorphisms of S 2 has a strong deformation retraction to the rotation group SO(3). Theorem B.1.5 ([Hat1]) Diff (S 3 ) is homotopy equivalent to O(4), where O(4) is the orthogonal group in dimension 4. Remark B.1.6 A generalized version of Smale’s conjecture for elliptic 3-manifolds containing incompressible Klein bottles and lens spaces L(p, q), where p ≥ 3, is proven in [HKMR]. Example B.1.7 Examples of the mapping class group of orientable surfaces. (1) Mod+ (S 2 ) = 1 [Deh3, Deh5, BC]1 This follows from Alexander’s trick, Theorem B.1.2. Let S 2 be defined in R3 as the unit 2-sphere {(x, y, z) ∈ R3 ; x 2 +y 2 +z 2 = 1}; then we can use Theorem B.1.2 to show that homeomorphisms fixed on the boundary of the disks {(x, y, z) ∈ S 2 ; z ≥ 0} and {(x, y, z) ∈ S 2 ; z ≤ 0} are isotopic to the identity. When we restrict to the plane z = 0, we have either an orientation-reversing homeomorphism, φ(x, y) = (x, −y), or the identity.  2 ) = Z [Deh3, Deh5, BC] (2) Mod+ (F0,2 2 A sketch of a proof can be found in [FaMa]. First construct a homomorphism ϕ  to Z via the action on the two punctures, from the mapping class group of F0,2 2

1 The proof was first published in [BC] with credit given to J. H. Roberts (most likely John Henderson Roberts, PhD, University of Texas at Austin 1929). 2 F0, n will denote the sphere with n punctures; the difference between a surface with punctures and a surface with boundary is that in the mapping class, the punctures are allowed to permute. See [FaMa] and [BC] for more details.

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and then prove that ϕ is an isomorphism. Surjectivity of ϕ is given; what needs  that to be shown is that an orientation-preserving self-homeomorphism on F0,2 preserves the punctures is isotopic to the identity. This can be shown by cutting the surface along an arc with endpoint equal to the two punctures and then applying Alexander’s trick, Theorem B.1.2. (2’) Mod+ (Ann) = Z [Deh3, Deh5] A proof can be found in [FaMa]; they proved this by constructing a homomorphism ϕ from the mapping class group of Ann to Z via the universal cover of Ann and then by proving that ϕ is an isomorphism. (3) Mod+ (T 2 ) = SL(2, Z) [Deh3, Deh5] A proof can be found in [FaMa]; they proved this by constructing a homomorphism ϕ from the mapping class group of T 2 to SL(2, Z) via the action on the first homology group of T 2 , H1 (T 2, Z)  Z2 ,3 and then proving that ϕ is an isomorphism.  ) = S where S is the permutation group on three elements [Deh3, (4) Mod+ (F0,3 3 3 Deh5, BC] A proof is given in [FaMa]; they proved this by constructing a homomorphism ϕ  to the permutation group S via the action from the mapping class group of F0,3 3 on the three punctures and then proving that ϕ is an isomorphism.

(4’) Mod+ (F0,3 ) = Z ⊗ Z ⊗ Z. [FoMa, FaMa] This can be derived from the generalization provided in (5’) below.  ) = PSL(2, Z)  (Z × Z ). [Deh4, Deh5, BC] (5) Mod+ (F0,4 2 2 A proof is given in [FaMa]; they proved this using hyperelliptic involutions and by constructing a homomorphism ϕ from the mapping class group of F0,4 to PSL(2, Z) as well as a right inverse of ϕ and then proving that ker ϕ  Z2 × Z2 .

(5’) Mod+ (F0,4 ) = PB3 ⊗ Z ⊗ Z ⊗ Z, where PB3 is the pure braid group on 3 strands. [FoMa] This follows from the generalization Mod+ (F0,n ) = PBn−1 ⊗ Zn−1 . A proof is given in [FoMa]; they proved this by viewing F0,n as a disk with n − 1 holes and constructing a homomorphism ϕ from the mapping class group of F0,n to PBn−1 and then constructing a right inverse and proving that ker ϕ  Zn−1 is generated by twists about the n − 1 holes which is in the center of Mod(F0,n ). Example B.1.8 Examples of the mapping class group of non-orientable surfaces. 1. Mod(Mb) = {1} where Mb is the Möbius band. [Eps] This follows from a theorem by David B. A. Epstein in [Eps] stated below:

3 The induced map is actually ϕ : Mod± (T 2 ) → GL(2, Z) which is the homomorphism used to prove that the extended mapping class group of T 2 is GL(2, Z) (This result was proven by Jakob Nielsen in his PhD thesis in [Nie]). However, in our example, we are only recognizing orientation-preserving homeomorphisms which allows us to restrict the range to SL(2, Z).

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Theorem B.1.9 If P is a Möbius band and F is a piecewise linear selfhomeomorphism which is fixed on the boundary, then F is piecewise linearly isotopic to the identity by an isotopy which is fixed on ∂P. 2. Mod(RP2 ) = {1} If we represent RP2 as S 2 /∼, where ∼ is the antipodal relation on S 2 , then we can apply the same argument as in the proof of Mod(S 2 ) = {1}. 3. Mod(Kb) = Z2 × Z2 . [Lic2] A proof can be found in [Lic2] where W. B. R. Lickorish shows that any homeomorphism h from Kb to itself is isotopic to one of the four homeomorphisms, idKb , f , c, or f ◦ c, where f and c commute and f 2 = c2 = idKb . A detailed description will be given at the end of this section. Definition B.1.10 ([Deh3, Lic8]) Let F be a compact oriented surface, c → F be a simple closed curve embedded in F, and Ann be an annular neighborhood of c in F. Then a Dehn twist is a homeomorphism hc : F → F defined by the following conditions: 1. The homeomorphism hc restricted to F\Ann is the identity, hc | F\Ann = id F . 2. The homeomorphism hc restricted to Ann, hc | Ann , is given by hc (eiθ , t) = (ei(θ−2πt), t) after parametrizing Ann as S 1 × [0, 1] in an orientation-preserving manner. Example B.1.11 A Dehn twist can be visualized as cutting the annulus at its outer boundary, twisting by 2π in a clockwise rotation, and identifying the boundary back in F. A smooth example of a Dehn twist in an annulus with a curve connected to a boundary point of each component of ∂Ann is given below.

.

Definition B.1.12 ([Lic8]) Let F be a compact oriented surface, and let p, q → F be oriented simple closed curves embedded in F; then p and q are called twist equivalent, denoted by p ∼c q, if there exists a sequence of Dehn twists hc1 , . . . , hck and an element n ∈ NF such that (n ◦ hc1 ◦ · · · ◦ hck )(p) = q. The next five results are due to Lickorish in [Lic1]; they are preliminary results to Theorem B.1.19 relating Dehn twists to orientation-preserving homeomorphisms of surfaces. Lemma B.1.13 The relation ∼c is an equivalence relation.

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Proof The reflexive property is trivial. The symmetry property holds because the inverse of a Dehn twist is also a Dehn twist. Assume p1 ∼c p2 by a sequence of Dehn twists hc = hc1 ◦ · · · ◦ hck and an element n ∈ NF and p2 ∼c p3 by a sequence of Dehn twists hc = hc1 ◦ · · · ◦ hck and an element n  ∈ NF . Then (n ◦ hc )(p1 ) = p2 and (n  ◦ hc )(p2 ) = p3 . Therefore, (n  ◦ n ◦ hc ◦ hc )(p1 ) = p3 which implies that  p1 ∼c p3 . Therefore, ∼c is transitive. Lemma B.1.14 Let F be a compact oriented surface, and let p, q → F be oriented simple closed curves embedded in F. Suppose p and q intersect transversely at one point; then p ∼c q. Proof Let c1 and c2 be simple closed curve that runs parallel to and slightly displaced from q and p, respectively. Let hc1 and hc2 be Dehn twists about c1 and c2 , respectively. Then (hc2 ◦ hc1 )(p) is isotopic to q, that is, there exists n ∈ Nx such that (n ◦ hc2 ◦ hc1 )(p) = q. An illustration is shown below:

A closer inspection of hc2 ◦ hc1 (p) is given below:

Therefore, p ∼c q.



Corollary B.1.15 Let F be a compact oriented surface, and let p1, p2, . . . , pr → F be oriented simple closed curves embedded in F, which do not separate F, and suppose that pi intersects pi+1 transversely at one point for i = 1, 2, . . . , r − 1. Then p1 ∼c pr . Proof By Lemma B.1.14, pi ∼c pi+1 . It follows that p1 ∼c pr since ∼c is transitive. Lemma B.1.16 Let F be a compact oriented surface, and let p, q → F be disjoint oriented simple closed curves embedded in F that do not separate F. Then p ∼c q. Proof There exists a simple closed curve r that intersects both p and q transversely at one point and does not split the surface. An example is depicted below where we consider a surface obtained by cutting F along p and q.

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Then by Corollary B.1.15, p ∼c q.

·· ·

·· ·

(a) Case 1: We start at point while moving along and arrive at point on the opposite side



·· ·

·· ·

(b) A simple closed curve is constructed from by running along between and . 1

·· ·

·· ·

(c) Case 2: We start at point while moving along and arrive at point on the same side.

·· ·

·· ·

2

(d) We construct two simple closed curves such that at least one does not separate .

Fig. B.1: The two cases that arise when you move along p from one intersection point, A, to the next intersection point, B. In case 1, we approach B from the opposite side, and in case 2, we approach B from the same side

Proposition B.1.17 Let F be a compact oriented surface, and let p, q → F be oriented simple closed curves embedded in F such that neither separates F. Then p ∼c q. Proof Let n be the number of intersection points between p and q. We will proceed with a proof by induction on n; Lemmas B.1.14 and B.1.16 serve as the base case. Assume n ≥ 2 and the proposition is true for intersection points less than n. Take

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two consecutive intersection points on p, say A and B. When we start from point A and run along p to point B, there are two cases to consider. Case 1. Suppose we arrive at B on the opposite side of q from where we started, as shown in Fig. B.1a. Let r be a simple closed curve constructed by starting slightly displaced from A, moving parallel to p until close to B, and then crossing q once between A and B, as shown in Fig. B.1b. Since r ∩ q has only one intersection point, then by Lemma B.1.14, r ∼c q. Since r ∩ p has less than n intersection points, then by the inductive hypothesis, r ∼c p. Finally, by Lemma B.1.15, we have p ∼c q. Case 2. Suppose we arrive at B on the same side of q from where we started, as shown in Fig. B.1c. Then we can construct two simple closed curves r1 and r2 by starting slightly displaced from A. For r1 , we start slightly above p, and for r2 , we start slightly below p, follow parallel to p, and then follow parallel to q, as shown in Fig. B.1d. This construction results in r1 and r2 disjoint from q. Notice that they follow a common curve between point A and point B and then they follow disjoint segments of q. Since q does not separate the surface, then at least one of the curves, r1 or r2 , does not separate the surface, call it r. Since r is constructed to be disjoint from q, then by Lemma B.1.16, r ∼c q. We also have that r ∩ p has less than n intersection points; by the inductive hypothesis, we have r ∼c p. Finally, by Lemma B.1.15, we  have our desired result, p ∼c q. A theorem proven in [Deh3, Deh5, Lic1] relates Dehn twists to orientationpreserving homeomorphisms of surfaces which naturally relates Dehn twists to the mapping class group of surfaces. The following is a lemma we will use to prove the theorem. Lemma B.1.18 ([Deh3, Deh5]) Any orientation-preserving homeomorphism h of a disk with n holes, which is fixed on the boundary, is isotopic to the product of a sequence of Dehn twists. Theorem B.1.19 (Dehn’s Theorem [Deh3, Deh5]) Any orientation-preserving homeomorphism h of a compact orientable surface F onto itself is isotopic to the product of a sequence of Dehn twists. In particular, isotopy classes of Dehn twists on a surface, F, generate the mapping class group of F. Proof The proof follows the proof in [FoMa]. By the classification theorem for surfaces, any compact oriented surface F is homeomorphic to the surface of a handlebody of a certain genus g with d disks removed. We choose g non-separating simple closed curves, a1, a2, . . . , g, that cut the surface into a disk with 2g − d − 1 holes. For each i = 1, 2, . . . , g, a homeomorphism h sends the non-separating curve ai to some other non-separating curve h(ai ). Our goal is to construct a sequence of Dehn twists that will return each curve back to their former places. We start with i = 1. Since a1 and h(a1 ) do not split the surface F, then by Proposition B.1.17, we have that there exists a sequence of Dehn twists, denoted by hc1 , such that hc1 ◦ h(a1 ) = a1 . Similarly, there exists a sequence of Dehn

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twists, denoted by hc2 , such that hc2 ◦ hc1 ◦ h(a2 ) = a2 and hc2 fixes hc1 ◦ h(a1 ). By applying this process for all i = 1, 2, . . . , g in sequential order, then we have that there exist sequences of Dehn twists, denoted by hc j , such that hc ◦ h(a j ) = hcg ◦hcg−1 ◦· · ·◦hc1 ◦h(a j ) = a j for j = 1, 2, . . . , g. This implies that h(a j ) = hc−1 (a j ). After cutting F along the curves a1, a2, . . . , ag , we obtain a disk with 2g + d − 1 holes. Therefore, hc ◦ h is a homeomorphism on the disk with 2g + d − 1 holes that  is fixed on the boundary. By Lemma B.1.18, we are done. Originally, for closed surfaces, Dehn proved that a finite number of curves suffice to generate the mapping class group, then Lickorish in [Lic1] improved this number to 3g − 1 curves on the surface, and later Stephen P. Humphries in [Hum] proved the result for 2g + 1 curves when g > 1 (see Fig. B.2). Humphries also proved that 2g + 1 is the smallest number of curves. James McCool in [McC] was the first to present an algorithm to defining the relations for the mapping class group of closed surfaces.

Fig. B.2: An illustration of the minimal number of curves needed to generate the mapping class group of a genus g > 1 closed surface

We will conclude this section with the relationship between Dehn twists and the mapping class group of non-orientable surfaces. Definition B.1.20 ([Lic2]) Let Y be a Möbius band with a disk removed and another Möbius band glued in its place. Then there exists a homeomorphism h : Y → Y which leaves ∂Y fixed and reverses the orientation of the orientation-reversing path around the glued Möbius band; in other words, it slides one crosscap once around the other. If Y is embedded in a non-orientable closed 2-manifold, X, then h induces h : X → X, a homeomorphism defined by h|Y = h and h|X\Y = id X . h is called a Y -homeomorphism. The Y -homeomorphism is also known as a crosscap slide; for more details, see [Lic2, Stu]. Remark B.1.21 The Y -homeomorphism can be rewritten as Y = hw ◦ u where hw is a c-homeomorphism about w (shown in Fig. 18.4) and u is a crosscap switch. Theorem B.1.22 ([Lic3]) Let X be a closed, connected, non-orientable 2-manifold such that X  RP2 , and let h : X → X be a homeomorphism. Then there exists a

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Y -homeomorphism y : X → X, such that h  hc ◦ y where hc is a product of Dehn twists and = 0 or 1. Corollary B.1.23 ([Lic2, Lic3]) Let X be a closed non-orientable surface and T (X) be the subgroup of Mod(X) generated by Dehn twists on X. Then T (X) is a subgroup of index 2 in the mapping class group, Mod(X). Recall Example B.1.8 Mod(Kb)  Z2 × Z2 , in particular, that any selfhomeomorphism on Kb is isotopic to one of the four homeomorphisms idKb, f , c, or, f ◦ c, where f and c commute and f 2 = c2 = idKb . We will now describe the homeomorphisms in terms of products of Y -homeomorphisms and Dehn twists. A Klein bottle can be considered as being a 2-sphere with two disjoint disks removed and replaced with two Möbius bands, M1 and M2 . The homeomorphism f is a Y homeomorphism described by sliding one crosscap about a curve intersecting both crosscaps. Lemma 1 in [Lic2] states that there are four nontrivial isotopy classes of simple closed orientation-preserving curves on the Klein bottle; the curves are shown in Lecture 18 , denoted by x, y, z, and w.4 As you can see in Fig. 18.4 , w intersects each crosscap. Lickorish in [Lic2] shows that Dehn twists on each of these curves are isotopic to the identity except for hw , a Dehn twist about the curve w. In our example, the homeomorphism c is isotopic to hw . Therefore, any self-homeomorphism on Kb is isotopic to one of the four homeomorphisms: the identity idKb , crosscap slide f , a Dehn twist hw , or the composition f ◦ hw which is a crosscap switch.

B.2 Integral and Dehn Surgery In 1910, M. Dehn in [Deh2] created a technique for constructing “Poincaré spaces” that is now known as a special case of Dehn surgery. This technique was generalized to what we now know as Dehn surgery in the work of R. H. Bing in the late 1950s [Bin1, Bin3]. Since then, there has been an explosion of results. In the 1960s, A. H. Wallace in [Wall] and W. B. R. Lickorish in [Lic1] proved that every closed orientable 3-manifold can be obtained from Dehn surgery along a link. In 1972, L. Moser in [Mos] determined all 3-manifolds obtained from Dehn surgery along torus knots. In 1978, R. Kirby in [Kir2] described an equivalence relation on surgery descriptions generated by two moves called Kirby moves. Then in the late 1980s and early 1990s, S. A. Bleiler and R. A. Litherland in [BlLi] and Wu in [Wu2] completely determined when surgeries on satellite knots produce lens spaces. In this part, we will introduce Dehn surgery and end with a proof of the Lickorish-Wallace theorem. Definition B.2.1 Let .K ⊂ M be a knot in a closed orientable 3-manifold M, that is, a smooth embedding of a simple closed curve into M. The tubular neighborhood of .K, denoted by .U (K), is an embedding of a solid torus, . D2 × S 1 , such that the core, 4 The curves are used in [BIMP1] to construct the Gram determinant of type Möbius band.

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{0} × S 1 , forms the knot, K. Similarly, the tubular neighborhood of a link . L ⊂ M is the tubular neighborhood of each component.

Definition B.2.2 We will discuss the three ways to represent the framing of a link that is used throughout the lectures. The following three definitions are equivalent in .S 3 = R3 ∪ {∞}. 1. Annular framing In the annular framing setting, we view a framed knot .K as an annulus embedded in .S 3 , .K → S 3 . The framing number of .K is defined as the linking number of the unoriented boundary of the annulus .S 1  S 1 after orienting both .S 1 in the same direction. A framed link .L with n components is a collection of n annuli embedded in .S 3 ; the framing number is the collection of framing numbers associated with each component. This definition is also valid for orientable 3manifolds; however, the notion of a framing number can no longer be defined (except for homology spheres). In this case, we can only discuss the change in framing; see Lecture 15. 2. Blackboard framing In the blackboard framing setting, we view a framed knot .K → R3 as a projection of the framed knot described in the annular framing setting onto .R2 where the normal vector for every point on the annulus is perpendicular to the plane. We view .K as a knot diagram with an associated framing number equal to the writhe K ). A framed link, .L , is viewed as a link diagram where of the knot diagram, .w(K each component .L i is a framed knot with framing number equal to the writhe, L i ). The framing number of .L is the collection of framing numbers associated .w(L with its components. This definition is also valid for .Fg,d × I; see Lecture 11. 3. Integral surgery framing In the integral surgery setting, we view a framed knot .K → S 3 as an unframed knot K accompanied with the framing number k that describes a curve in the boundary of the tubular neighborhood, .c ⊂ ∂U(K), that twists k times around the meridian and once around the preferred canonical longitude of .∂U(K). The framing number is equal to the linking number of K and c where they are oriented in the same direction. A framed link is then viewed as a collection of framed knots described above. This definition is also valid for integral homology spheres. Definition B.2.3 Let .K → M be a knot embedded in a closed, orientable 3-manifold M. Then the manifold . XK = M\int(U(K)) obtained by removing the interior of the tubular neighborhood of K from M is called the knot exterior. Lemma B.2.4 Let K be the unknot and . M = S 3 ; then . XK is homeomorphic to a solid torus. Proof Recall that .S 3 can be decomposed into the Heegaard decomposition of genus 1, that is, two solid tori .V1 = D2 × S 1 and .V2 = S 1 × D2 where

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S 3 = ∂D4 = ∂(D2 × D2 ) = (D2 × ∂D2 ) ∪ (∂D2 × D2 ) = (D2 × S 1 ) ∪ (S 1 × D2 ).

We can deform the unknot K via ambient isotopy in .S 3 to be the core of .V1 such that U (K) = V1 ; then .V2 = S 3 \int(U(K)). 

.

Definition B.2.5 Integral surgery on a knot .K in an orientable 3-manifold, M, is composed of two steps, Dehn drilling and Dehn filling. Dehn drilling is the process of taking the tubular neighborhood of the underlying unframed knot .K, .U (K), which is homeomorphic to . D2 ×S 1 , and removing it from M so that we obtain two manifolds: 2 1 . XK = M\int(U(K)) and .V = U(K)  D × S . Dehn filling is the process of using an orientation-reversing homeomorphism .ϕ : ∂V → ∂ XK determined by the curve in .∂V given by the framing of the knot .K to add V back into . XK , that is, adding a 2-handle along the framing, and then cap it off with a 3-ball. The result . XK ∪ϕ V is a closed orientable manifold; we say this manifold is obtained by integral surgery along .K from M. A special case of integral surgery is when . M = S 3 ; in this case, we will denote by . MK the manifold obtained by integral surgery on the framed knot .K from .S 3 . We may define integral surgery along a framed link by applying the process to each component. The manifold obtained by integral surgery along a framed link from .S 3 is denoted by . ML . Since .∂V  ∂ X  T 2 , then .ϕ : T 2 → T 2 is an orientation-reversing selfhomeomorphism. Since .Mod(T 2 )  SL(2, Z), then the induced homomorphism .ϕ∗ : π1 (T 2 ) → π1 (T 2 ) is defined by . Aτ where . A ∈ SL(2, Z) and .τ is a .2 × 2 matrix such that .det(τ) = −1. Therefore, we can define .ϕ∗ to be        q s −1 0 −q s −1 0 . ϕ∗ = = where τ = and qr + ps = 1 for q, p, r, s ∈ Z. −p r 0 1 p r 0 1 Recall that .π1 (T 2 )  Z⊕Z. Let .{μ, λ  } be basis elements of .π1 (T 2 ); they represent the meridian and preferred canonical longitude (the longitude that lies on the Seifert surface of K) of .∂V. Similarly, let .{μ, λ} represent the meridian and preferred canonical longitude of .∂ X. Then .ϕ∗ (μ) = −qμ + pλ, and . X ∪ϕ V is completely determined by .ϕ∗ (μ) which is a curve in .∂V. As mentioned before, the attaching homeomorphism .ϕ is determined by a simple closed curve c in .∂V that represents the framing of the knot .K . By construction, we have that . μ corresponds to .λ and also     .λ corresponds to . μ in . π1 (∂V), which implies that .c = ϕ∗ (μ ) = pμ − qλ ∈ π1 (∂V). 3 For . M = S , the framing number is determined by .−p/q; therefore, .q = ∓1, and .±p is the framing number of the knot .K . Dehn filling is performed on a manifold M with toroidal boundary (possibly with a disjoint union of toroidal boundary); the orientation-reversing homeomorphism that glues a torus into M along a toroidal boundary, say .∂i M = T 2 , is determined by a curve in .∂i M. Furthermore, this curve does not necessarily need to correspond to a framing of a knot. In terms of our previous description, we have . M = X, and the gluing homeomorphism is completely determined by .ϕ∗ (μ) = −qμ + pλ where . {μ, λ} represent the meridian and preferred canonical longitude of . ∂ X = ∂i M.

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The curve representing the Dehn filling in M has slope .−p/q for . p, q ∈ Z and gcd(p, q) = 1. For examples, more details, and the applications of Dehn fillings, we refer the reader to Jessica Purcell’s book [Pur].

.

By Lemma B.2.4, integral surgery along a framed unknot in .S 3 yields a manifold with Heegaard splitting of genus 1. That is, if .K is a framed unknot, then . MK = H ∪ϕ H  where H and .H  are handlebodies of genus 1 and .ϕ is an orientationreversing homeomorphism between .∂H and .∂H . By applying the construction of lens spaces explained in Appendix A, we have the following examples. where is the trivial knot with framing 1. Example B.2.6 Here, we have where .H = XK and .H  = V are handlebodies of genus 1. The homeomorphism is determined by .ϕ∗ (μ) = λ− μ ∈ π1 (∂H). Therefore,

Example B.2.7

, where

is the trivial knot with framing 0.

where .H = XK and .H  = V are In this example, we also have handlebodies of genus 1. In this case, the homeomorphism is determined by .ϕ∗ (μ) = −μ ∈ π1 (∂H). Therefore,

Fig. B.3: Poincaré homology sphere from integral surgery

Example B.2.8 . ML is homeomorphic to the Poincaré homology sphere .Σ(2, 3, 5) where .L is the right-handed trefoil with framing number equal to one, shown in Fig. B.3a. The Poincaré homology sphere is defined by Seifert manifolds (see Appendix A Definition A.7.1). Recall that .π1 (XL ) < x, y; xyx = yxy >; this implies that .π1 (ML ) < x, y; xyx = yxy, ϕ∗ (μ) = e >. In this example, .ϕ(μ) = m + l, where m and l are shown in Fig. B.3b. Therefore, the relation .ϕ(μ) = e becomes −3 ) = e in . π (M ) where . x, y, and z are generators shown in Fig. B.3b and .(x)(zxyx 1 L −1 .z = x yx ∈ π(ML ). This yields the relation . x 2 yx 2 y = x 5 . Since . xyx = yxy, then 2 2 2 3 2 5 . x yx y = xyxyxy. Let .a = xy; then we have .(xa) = a and .(xa) = x . Therefore, 2 3 5 . π(ML ) < a, x; (xa) = a = x > π1 (Σ(2, 3, 5)).

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We have .H1 (XL ) < m >, which implies that .H1 (ML ) < [m] H ; [ϕ(μ)] H = 0 >. We also have .[ϕ(μ)] H = [m + l] H = [m] H ; therefore, .H1 (ML ) = 0. This tells us that . ML is an integral homology sphere.We will now show the Heegaard diagram of . ML , which is an approach to the Poincaré homology sphere. First take the tubular neighborhood of the link, .U (L), which is a handle body of genus 1, and include the curve c defining the framing of .L . Then attach a handle to .U (L), with meridian . μ, in order to obtain a handle body of genus 2. Next, we want to show that .S 3 \int(U(L) ∪ (1 − handle)) is also a handle body. This is done in Fig. B.4 where we show a series of isotopy deformations performed on the handle body, .U (L) ∪ (1 − handle), which leads to the Heegaard diagram of . ML . Notice that it is equivalent to the Poincaré homology sphere’s Heegaard diagram. Notice that for integral surgery, we were restricted to .q = ±1, but as seen in Dehn fillings, the homeomorphism still makes sense for any .q ∈ Z where .gcd(p, q) = 1. The reason why we were restricted to .q = ±1 is because the framing of a knot is obtained from the curve c which wraps around the longitude of .U (K) once and the meridian p times. Dehn surgery (also known as rational surgery when . M = S 3 ) is a generalized version of integral surgery where we allow any orientable compact manifold M (or any .q ∈ Z where .gcd(p, q) = 1 when . M = S 3 ). Definition B.2.9 Let M be an orientable compact 3-manifold and K be a knot embedded in M. Dehn surgery is composed of two steps, Dehn drilling and Dehn filling. First perform a Dehn drilling on M by removing the tubular neighborhood of K, .U (K) from M, and then perform a Dehn filling by using an orientationreversing homeomorphism .ϕ determined by a curve, c, in .∂U(K). As before, the result . XK ∪ϕ U(K) is an orientable manifold. For . M = S 3 , we may use the same induced homomorphism and the same basis elements for .π1 (∂U(K)) and .π1 (∂ XK ) as described in Definition B.2.5. The curve is then described by the rational number  .−p/q where .c = ϕ∗ (μ ) = pλ − qμ ∈ π(∂ XK ). We consider c to be in . ∂U(K); in this   case, .c = pμ − qλ ∈ π1 (∂U(K)). This process is also known as rational surgery on a knot with a given rational number .−p/q (usually displayed in the upper right-hand corner) to describe the orientation-reversing homeomorphism .ϕ. We will denote the manifold obtained by Dehn surgery along .K from .S 3 by . MK where .K is a knot with a given rational number. Example B.2.10 (Lens Spaces) . MK  L(p, q), where Dehn surgery from .S 3 along the unknot with rational number .−p/q yields a manifold with Heegaard splitting of genus 1. . MK = H ∪ϕ H  where H and .H  are handlebodies of genus 1 and .ϕ is described by .ϕ∗ (μ) = pλ − qμ ∈ π1 (XK ). By applying the construction of lens spaces explained in Appendix A, then it is clear that this construction implies that . MK  L(p, q).

Example B.2.11 Let .K 1 and .K 2 be the two components of .L with framing number .−p and .−q,

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−2

(a)

−2

−2

( ).

(b)

( ‰ 1 − handle).

(c) Isotopy deformation.

−2 (d) Isotopy deformation.

3 4

2

1 5

2 1 7 3 6 4 5

(e) The complement is also a handlebody.

1 5

2

3 4

1 2 3 7 4 6 5

(f) Heegaard diagram of

Fig. B.4: Heegaard diagram of . ML where .L is the left-handed trefoil with framing number equal to 1

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respectively. Since .K1 and .K2 are both unknots, then by removing the tubular neighborhood of the underlying link L from .S 3 , we obtain a toric shell 3 2 . S \int(U(L))  T × [0, 1]. Let .V1 = U(K1 ) and .V2 = U(K2 ); then we will use   −1 0 2 . ϕ1 : ∂V1 → T × {0} defined by the induced homomorphism . ϕ1 ∗ = to add p 1 2 2 .V1 back into .T × [0, 1]. Similarly we use . ϕ2 : ∂V1 → T × {1} and its induced   −1 0 to add .V2 back into .T 2 × [0, 1]. We can use this to homomorphism .ϕ2 ∗ = q 1 define a homeomorphism .ξ : ∂V2 → ∂V1 ; its induced homomorphism is defined by         −1 0 0 1 −1 0 −q −1 −1 0 −1 .ξ∗ = ϕ1 ∗ = . This construction ϕ2 ∗ = p 1 10 q 1 pq − 1 p q 1 yields a lens space, in particular . L(pq − 1, q). A generalization of Example B.2.10 is given below. Theorem B.2.12 Let .a1, a2, . . . , an be integers and − =

1

−

2

−

−

3

...

−1

− ,

.

then . ML  L(p, q) if p and q are coprime and determined by the continued fraction decomposition 1 p = an − . . q 1 an−1 − 1 .. . − a2 − a11 Lemma B.2.13 Let . M1 be a closed orientable 3-manifold, and consider the Heegaard splitting . M1 = H ∪h1 H  where H and .H  are two handlebodies of genus g. Let . h2 : ∂H → ∂H  be a homeomorphism of the surface of two handlebodies such that . h1 = h2 ◦ (hc1 ◦ · · · ◦ hcn ) where . hci is a Dehn twist about a simple closed curve .ci ⊂ ∂H. Then a closed orientable 3-manifold . M2 with Heegaard splitting   . M2 = H ∪h2 H is obtained from . H ∪h1 H by integral surgery along a link . L ⊂ M1 where each component . Li is isotopic to .ci . Proof Following Lickorish’s proof in [Lic1], we start by first considering .n = 1. Push the curve .c1 inside of the handlebody H to obtain a knot . L1 in H. Consider the tubular neighborhood of . L1 , .U (L1 ), and let . μ denote the meridian of .∂U(L1 ) and A denote an annulus in H whose boundary components are .c1 and a longitudinal curve .λ of . ∂U(L1 ). Let . h c1 : H − int(U(L1 )) → H − int(U(L1 )) be defined by cutting along A and . ∂U(L1 ) to remove .int(U(L1 )) and then twisting along .c1 and gluing back along A;

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this process results in a Dehn twist on .∂H along .c1 and a Dehn twist on .∂U(L1 ) along .λ at the expense of cutting out a solid torus from H and having to glue it back in, that is, Dehn surgery along . L1 in . M1 (see Fig. B.5). 1 = (H − int(U(L1 ))) ∪h1 H  and . M 2 = (H − int(U(L1 ))) ∪h2 H , and Consider . M 1 → M 2 by define a homeomorphism .ϕ : M  .

ϕ(x) =

h c1 (x), if x ∈ H − int(U(L1 )) x, if x ∈ H  .

This homeomorphism is well defined since .ϕ|∂H = hc1 and . h1 = h2 ◦ hc1 . This implies that after removing the solid torus .U (L1 ) from . M1 and . M2 , the manifolds become homeomorphic. Therefore, . M2 is obtained from Dehn surgery along . L1 in . M2 . Furthermore, . ϕ maps . μ to . μ ± λ which implies that the surgery is an integral surgery. We can repeat this process for other Dehn twists . hci by arranging the solid tori {U(Li )} to be disjoint. The composition of a sequence of Dehn twists then yields a sequence of integral surgeries along the components of a link L where each  component . Li is isotopic to .ci .

.

Fig. B.5: An illustration of cutting along A and .∂U(L1 ) to remove .int(U(L1 )), twisting along .c1 , and gluing back along A

Theorem B.2.14 (Lickorish-Wallace Theorem [Lic1, Wall]) Every closed orientable 3-manifold can be obtained from .S 3 by performing integral surgery on a framed link. Proof We follow Lickorish’s proof in [Lic1]. By Theorem A.4.11, every 3-manifold has a Heegaard splitting of genus g, that is, every 3-manifold can be decomposed into   . M = H ∪h H where H and . H are handlebodies of genus g and h is an orientationreversing homeomorphism on .∂H = ∂H . Consider the Heegaard splitting of .S 3 , 3  . S = H ∪h1 H , and let M be an arbitrary closed orientable 3-manifold with Heegaard

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splitting . M = H ∪h2 H . Let . h = h2−1 ◦ h1 ; then h is an orientation-preserving selfhomeomorphism on the surface .∂H = ∂H . By Theorem B.1.19, h is a product of Dehn twists, that is, . h = hc1 ◦ · · · ◦ hcn for some n where .ci is a curve in .∂H for .1 ≤ i ≤ n. Therefore, . h1 = h2 ◦ hc1 ◦ · · · ◦ hc n , and by Lemma B.2.13, M is obtained  by integral surgery along a link L from .S 3 .

B.3 Kirby Calculus The representation of a closed connected orientable 3-manifold by integral surgery in .S 3 is not unique, as indicated in Examples B.2.10 and B.2.11. In 1978, Robion C. Kirby in [Kir2] described an equivalence relation on integral surgery in .S 3 that is generated by two moves called Kirby moves. As Kirby stated in [Gos], while attending a conference in Tokyo in the spring of 1973, he overheard Takao Matumoto mention the Kummer surface numerous times (in the 1970s, K3 was called the Kummer surface), and his interest in K3 and 4-manifolds led him to the Kirby calculus. In this section, we will present the two elementary moves (Kirby moves) on a link in 3 . S that does not change the 3-manifold. Definition B.3.1 Kirby 1 move. Kirby 1 move will be denoted by K1. This move adds or deletes a disjoint unknot with framing .±1 to/from the link .L , .

Kirby 2 move. Kirby 2 move will be denoted by K2. This move involves two link components, where one component is slid over the other. Let .L i and .L j be link components of .L where .L = L  ∪ L i ∪ L j , let .c j be the curve in .∂U(L j ) as described in Definition B.2.2 (3), and let .c˜j be a copy of .c j pushed away from .L j ; then K2

.

L  ∪ L i ∪ L j −−−−−→ L  ∪ L # ∪ L j ,

where the underlying unframed link, . L# , is obtained by taking the connected sum of Li and .c˜j by some band b disjoint from the rest of L, . L# = Li #b c˜j . The framing number of .L # is then equal to .n# = ni + n j + 2lk(Li, L j ), where the sign of the linking number is determined by the chosen band b. If we are in the blackboard framing setting, then the moves will automatically keep track of the change in framing number.

.

Theorem B.3.2 ([Kir2]) Given two framed links .L 1 and .L 2 , then . ML 1  ML 2 if and only if we can obtain .L 2 from .L 1 by a sequence of Kirby moves, K1 and K2. Example B.3.3 Let .L be a framed link of two components shown below:

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.

We will perform K2 by sliding . L1 over . L2 in two different ways (via two different bands). First we produce .c˜2 , a copy of .c2 that is pushed away from .L 2 , and then take the connected sum of . L1 and .c˜2 via two different bands, .b1 and .b2 , as shown below:

.

The framing of .L # is determined by the connecting band; for .b1 , start by orienting . L1 , and then orient . L2 to match the orientation of . L1 #b1 c˜2 , as shown in the righthand side of Fig. B.6. The result gives .lk(L1, L2 ) = −2. Therefore, applying the K2 move using band .b1 yields a framed link .L # with the underlying unframed link . L1 #b1 c˜2 and framing number .n# = −3 + 1 + 2(−2) = −6. Similarly, for . b2 , we obtain .lk(L1, L2 ) = 2; see the left-hand side of Fig. B.6. Therefore, applying the K2 move using band .b2 yields framed link .L # with the underlying unframed link . L1 #b2 c˜2 and framing number .n# = −3 + 1 + 2(2) = 2.

Fig. B.6: Orienting . L1 and . L2 to solve for .n# = n1 + n2 + 2lk(L1, L2 ) where .n1 = 3 and .n2 = 1

Remark B.3.4 1. For a framed link .L , R. Kirby in [Kir2] refers to . ML as the 4-manifold obtained L . The notation . ML by adding 2-handles to the 4-ball . B4 corresponding to .∂L in this collection of lectures is .∂ ML in [Kir2]. We will denote the 4-manifold associated with . ML by .WL , that is, .∂WL = ML . 2. A Kirby 1 move corresponds to taking a connected sum of . ML and .S 3 .

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3. A Kirby 2 move corresponds to handle sliding in .WL , that is, sliding the .i th handle over the . j th handle through the band b. Example B.3.5 ([FR]) Let .L = L  ∪ L 0 ∪ L 1 ∪ L 2 be a framed link in .S 3 that contains an unknot .L 0 with framing number zero, a link component .L 2 disjoint from .L 0 , and a sublink .L 1 that is disjoint from .L  and linked by .L 2 and .L 0 , as shown in Fig. B.7. Then we may unlink .L 2 from .L 1 by a Kirby 2 move without changing the framing of . L2 .

Fig. B.7: One link component .L 2 ⊂ L can be unlinked from a sublink .L 1 ⊂ L by a Kirby 2 move and ambient isotopy if the sublink .L 1 is linked by an unknot with framing zero, .L 0

Exercise B.3.6 If .L 1 is an unknot in Example 1.39, show that . ML  ML  ∪LL 2 (i.e., we can remove .L 0 ∪ L 1 ). Hint: Notice that the lens space . L(1, 1) is homeomorphic to .S 3 . (See Appendix A.) Example B.3.5 and Exercise B.3.6 are special cases of Kirby’s theorem, stated below, which is only true for links in .S 3 . Theorem B.3.7 ([Kir2]) Let .L = L  ∪ L 0 ∪ L 1 ∪ L 2 be a framed link in .S 3 with the same conditions as in Example B.3.5. Then . ML  ML  ∪LL 2 ; equivalently, we can obtain .L  ∪ L 2 from .L by a sequence of Kirby moves. Definition B.3.8 (Blow-Up/Blow-Down Operation) 5 A blow-up operation, denoted by .K+ , is the following sequence of Kirby moves performed on a framed link .L :

.

5 In [FR], the blow-up and blow-down operations are called k-moves, while K1 moves are called special k-moves, and K2 moves are called .β-moves.

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A blow-up operation, denoted by .K− , is the following sequence of Kirby moves performed on a framed link .L :

.

A blow-down operation is the inverse of a blow-up, that is, if .L contains an unknot with .±1 framing, then we can use the reverse sequence of Kirby moves shown above; we will denote it by .K+−1 and .K−−1 , respectively. Roger Fenn and Colin Rourke in [FR] refined the Kirby calculus by showing that links related by blow-up and blow-down operations are sufficient for surgery in 3 . S . The authors also explained how to further extend the Kirby calculus to closed compact 3-manifolds by including an extra move. This extra move, denoted by K0 and defined below, was remarked by Kirby in [Kir2] as a necessary move to extend the Kirby calculus to surgery on arbitrary orientable 3-manifolds. Definition B.3.9 The Kirby 0 move,6 denoted by K0, removes or adds a disjoint union of a link component, .L 1 , with a meridional curve with zero framing, .L 0 ,

.

Theorem B.3.10 (Fenn-Rourke Theorem [FR]) Given two framed links .L 1 and .L 2 in .S 3 , then . ML 1  ML 2 if and only if we can obtain .L 2 from .L 1 by a sequence of blow-up and blow-down operations. Theorem B.3.11 ([FR]) Given two framed links .L 1 and .L 2 in a compact oriented 3-manifold M, and suppose . ML i is obtained from surgery on .L i in M, for .i = 1, 2, then . ML 1  ML 2 if and only if we can obtain .L 2 from .L 1 by a sequence of K1, K2, and K0 moves. Finally, Justin D. Roberts proved that the three moves K1, K2, and K0 are sufficient to extend the Kirby calculus to compact, connected, orientable 3-manifolds with boundary.

6 Note that from Theorem B.3.7, this move (also known as the circumcision move) is a result of a sequence of K1 and K2 moves for links in .S 3 .

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Theorem B.3.12 ([Rob2]) Given two framed links .L 1 and .L 2 in a compact, connected, orientable 3-manifold M with boundary, and suppose . ML i is obtained from surgery on .L i in M, for .i = 1, 2, then . ML 1  ML 2 if and only if we can obtain .L 2 from .L 1 by a sequence of K1, K2, and K0 moves.

B.4 The Linking Matrix Definition B.4.1 Let L be an oriented framed link with k components in S 3 , and let ni denote the framing number of the ith component. Then the linking matrix, denoted by AL = (ai j ), is a k × k matrix whose i jth entry is equal to  for i = j, ni .ai j = lk(Li, L j ) for i  j. Remark B.4.2 The linking matrix is symmetric since ai j = lk(Li, L j ) = lk(L j , Li ) = a ji . The link L from ML is not oriented, so the associated linking matrix depends on the choice of orientation for each component. Furthermore, the linking matrix is not an invariant of 3-manifolds. However, the number of positive and negative eigenvalues of the linking matrix will be used in Lecture 16 to construct an invariant of 3-manifolds. The affects of the Kirby moves on the linking matrix are as follows: Let L be an oriented framed link with k components and AL be the associated linking matrix. Kirby 1 move. If L is changed by a Kirby 1 move .

then the associated linking matrix changes from a k × k matrix into a matrix direct sum of AL and (±1) which is a (k + 1) × (k + 1) matrix, 0

AL .   ..  . AL − −−−−−→  0 . . . 1  K1+

0

AL .   ..  . AL −−−−−−→  0 . . .−1  K1−

Kirby 2 move. Suppose we slide L i over L j to produce the pair L # ∪ L j in L by using band b such that the band corresponds to the chosen orientation of the components. Then the new linking matrix is obtained from AL by adding the jth row to the ith row and then adding the jth column to the ith column. Suppose instead we used a band b, which does not correspond to the chosen orientation, to produce the pair L # ∪ L j in L . Then the new linking matrix is obtained

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from AL by subtracting the jth row from the ith row and then subtracting the jth column from the ith column. L ) and b− (L L ) the Definition B.4.3 Let L be an oriented framed link. Denote by b+ (L number of positive and negative eigenvalues of the linking matrix AL , respectively. L ), is the difference of Then the signature of the linking matrix AL , denoted by σ(L the number of positive and negative eigenvalues of AL . That is, .

L ) = b+ (L L ) − b− (L L ). σ(L

Let L be an oriented framed link with k components and AL be the associated linking matrix. Then the signature of the linking matrix changes under Kirby moves in the following way: Kirby 1 move. If L is changed by a Kirby 1 move in the following way: .

then the resulting linking matrix is equal to AL ⊕ (1). This operation increases the number of positive eigenvalues by one, . The result increases the signature by one, . Similarly, if L is changed by a Kirby 1 move in the following way: .

then the resulting linking matrix is equal to AL ⊕ (−1). This operation increases the number of negative eigenvalues by one, . The result decreases the signature by one, Kirby 2 move. Let L  be the resulting link after a K2 move on L . Recall that AL is symmetric and K2 performs an elementary row operation on the ith row from the jth row and an elementary column operation on the ith column from the jth column. Let F be a Frobenius matrix such that F AL produces a matrix from AL by adding (or subtracting) the jth row to the ith row. Then the affects of K2 on L change the associated matrix by the following operation: K2

.

AL −−−−−→ F AL F T .

By Sylvester’s law of inertia, the number of positive and negative eigenvalues remain L ) = σ(L L ). unchanged. Therefore, σ(L L ) in Lecture 16 in order to prove the Remark B.4.4 We will use the signature σ(L invariance of the Witten-Reshetikhin-Turaev invariant of 3-manifolds. We end this section on a result about even surgery. We will discuss its connections to 4-manifolds and parallelizable 4-manifolds in the next lecture.

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n be a framed link in S 3 . Then L is called an even K i }i=1 Definition B.4.5 Let L = {K link if the framing number of K i is even for i = 1, 2, . . . , n.

Theorem B.4.6 ([Mil2]) Every closed orientable 3-manifold can be obtained from S 3 by performing integral surgery along an even link. A simplified proof than that of [Mil2] can be found in [Sav] and [FoMa] which uses the linking matrix and characteristic sublinks.

B.5 Parallelizable Manifolds In this section, we give an introduction to bundles and tangent bundles with an abundance of classical examples and theorems as a means to set up a discussion about stably parallelizable and parallelizable manifolds.

B.5.1 Bundles and Tangent Bundles There are many definitions of bundles; however, we will work with locally trivial fiber bundles, that is, a direct generalization of a covering in which case the fiber is a discrete set. Definition B.5.1 A bundle over a topological space B is a triple .(E, π, B), denoted by π E −−→ B, where E is a topological space called the total space, B is also a topological space called the base space, and .π is a continuous surjective map .π : E → B called the projection. Let . p ∈ B; then the preimage of p under .π, denoted by .π −1 ({p}), is called the fiber at the point p. If for all . p ∈ B, .π −1 ({p})  F for some topological π π space F, then .E −−→ B is called a bundle over B with typical fiber F. Let .E −−→ B be a bundle with typical fiber F; then the bundle is called a fiber bundle if it is locally trivial, that is, if for every . p ∈ B, there exists an open neighborhood .Up of p such that .ϕ p : π −1 (Up ) → Up × F is a homeomorphism, called a local trivialization, making the following diagram commute.

.

−1 (

)

× Proj

π1

π2

Definition B.5.2 Let .E1 −−−→ B1 and .E2 −−−→ B2 be two bundles. A bundle morphism is a pair of continuous maps .( f , g) such that . f : E1 → E2, g : B1 → B2 , and the following diagram commutes:

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2 1

2

1

2

The notion of a bundle comes from the product; as a first example, we will construct the trivial bundle (or product bundle). However, in the rest of the examples to follow, we will see that bundles are a generalization of the product. Example B.5.3 Let M and F be topological manifolds and .E = M × F be equipped with the product topology of M and F (E is called the product manifold7 ). Define the projection .π : M × F → M by .(p, f ) → p which is surjective. Since E is π equipped with the product topology, then .π is also continuous; therefore, .E −−→ M is a bundle. Notice that .π −1 ({p}) = {p} × F  F for all . p ∈ M. A classical example of a bundle that is a generalization of the product is the bundle over .S 1 with typical fiber . I = [−1, 1] and total space the Möbius band, M b. Example B.5.4 (The Möbius Band) Let . I = [−1, 1]; one model of the Möbius band is given by taking . I × I and identifying the boundary lines .{−1} × I and .{1} × I together in opposite orientation. That is, . M b = I × I/({−1} × {t} ∼ {1} × {−t} for t ∈ I). We can see .S 1 in the model at . I × {0} since .{−1} × {0} is identified with .{1} × {0}. 1

.

Let .E = M b be a Möbius band, . M = S 1 , and .π : E → M be the projection defined by mapping all points down the middle line . I × {0} in the model, that is, π . π is defined by .(s, t) − −→ (s, 0). Then .π −1 ({(p, 0)}) = {p} × I  I for all . p ∈ S 1 . π Therefore, . M b −−→ S 1 is a bundle over .S 1 , but it is not the trivial bundle over .S 1 with typical fiber I because . M b  I × S 1 . π

Definition B.5.5 Let .E −−→ M be a bundle and .σ : M → E be a map such that . π ◦ σ = id M . Then .σ is called a section of the bundle. For any element . p ∈ M, .σ(p) is an element of the fiber at the point p, .σ(p) ∈ π −1 ({p}). In the next example, we will construct a section from the trivial bundle constructed in Example B.5.3.

7 See [Tu] for a thorough discussion on the product manifold.

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π

Example B.5.6 Let .E −−→ M be a bundle such that .E = M × F is the product π manifold and .π is the projection on the first coordinate, that is, .(m, f ) −−→ m. Then the fiber for any point . p ∈ M is homeomorphic to F. In this case, we can define a section .σ : M → M × F point-wise by . p → (p, s(p)) for any map .s : M → F. A vector bundle is a special fiber bundle that plays an important role in geometry and topology. We will be particularly interested in a special type of vector bundle over smooth manifolds called tangent bundles. Definition B.5.7 A vector bundle is a fiber bundle such that the fiber, denoted by V, is a vector space and for each local trivialization .ϕ p : π −1 (Up ) → Up × V, we have the following: For every .q ∈ Up , .

ϕ p |π −1 ({q }) : π −1 ({q}) → {q} × V

(B.1)

is a vector space isomorphism. Furthermore, the rank of the vector bundle is the dimension of its fiber. An example of a product bundle that is also a vector bundle (a trivial vector bundle) that will be used in the next section is called a trivial line bundle. π

Example B.5.8 A trivial line bundle over a manifold M is . M × R −−→ M where . π : M × R → M is defined by .(p, x) → p. π1

π2

Definition B.5.9 Consider two vector bundles .E1 −−−→ B and .E2 −−−→ B. Then a morphism of vector bundles with the same base space B is a continuous map . ϕ : E1 → E2 such that the following diagram commutes: 1

2 1 2

and .ϕ restricted to each fiber is a linear map. π

In particular, a vector bundle .E −−→ M over a smooth n-manifold is called trivial if .E φ M × V where .φ : M × V → E is a diffeomorphism such that . π ◦ φ : M × V → M is the projection on the first coordinate and . φ restricted to each fiber is an isomorphism of vector spaces. In the next example, we will use a smooth embedding of .S n into .Rn+1 with standard smooth structure to define the normal bundle to .S n in .Rn+1 , and then we will show that this is a trivial line bundle. This example will be important to the next section when we discuss stably parallelizable manifolds.

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Example B.5.10 Consider .S n := {x ∈ Rn+1 ; x ·x = 1}, and define the normal bundle π of .S n into .Rn+1 to be a vector bundle .NSn −−→ S n where the total space is defined by n n n+1 .NS = {( x, tx ) ∈ S × R ; t ∈ R} with projection map defined by .π(x, tx ) = x . For p}) = {( p, t p); t ∈ R}  R. We will show that this each point . p ∈ S n , we have .π −1 ({ bundle is a trivial line bundle by considering the diffeomorphism .ψ : NSn → S n × R defined by .( p, t p) → ( p, p · (t p)). Since .π ◦ ψ −1 ( p, t) = π( p, t p) = p, then it suffices to show that .ψ restricted to each fiber is a vector space isomorphism. Since for each  ∈ S n , .ψ|π −1 ({ p }) is defined by .( .p p, t p) → ( p, p · (t p)) for a fixed . p ∈ S n and the dot product is bilinear, then we have our desired result. Throughout the rest of this subsection, we will let M be a smooth orientable n-manifold. A tangent bundle is a special type of vector bundle; it plays an important role in discussing smooth maps between manifolds and framings of manifolds. In this part, we will focus on the sections of the tangent bundle and how these play a role in telling whether or not a tangent bundle is trivial, in other words, that a manifold is parallelizable. See [MiSt, Mil4, Tu] for more details about tangent bundles. Definition B.5.11 Let .γ : R → M be a smooth curve through a point . p ∈ M. Without loss of generality, we may assume that .γ(0) = p. Then the directional derivative operator at the point p along the curve .γ is the linear map . Xγ, p : C ∞ (M) → R defined by . Xγ, p ( f ) := ( f ◦ γ)(0). The directional derivative operator . Xγ, p is also known as a tangent vector at the point p. Definition B.5.12 Let M a smooth n-manifold, and let . p ∈ M; then the tangent space of M at p, denoted by .Tp M, is the set of all tangent vectors at p. That is, .Tp M = {Xγ, p ; γ is a smooth curve through p} and is equipped with .+ and .· where .+ : Tp M ×Tp M → Tp M is defined point-wise by .(Xγ, p + Xδ, p )( f ) := Xγ, p ( f ) + Xδ, p ( f ) and .· : R × Tp M → Tp M is defined by .(λ · Xγ, p )( f ) := λXγ, p ( f ). In order to show that .+ : Tp M × Tp M → Tp M is well defined, we must show that there exists a curve .α such that .(Xγ, p + Xδ, p ) = Xα, p . Let .(U, ϕ) be a chart about the point p, and then define .α : R → M by .α(t) = ϕ−1 ((ϕ ◦ γ)(t) + (ϕ ◦ δ)(t) − (ϕ ◦ γ)(0)). Then .α(0) = ϕ−1 ◦ (ϕ ◦ δ)(0) = p, and since .( f ◦ ϕ−1 ) : Rn → R, then by chain rule, we have .

Xα, p = ( f ◦ α)(0)  ∂( f ◦ ϕ−1 ) d(ϕ ◦ α)i d(( f ◦ ϕ−1 ) ◦ (ϕ ◦ α)) (0) = (0) (ϕ ◦ α(0)) i dt dt ∂ϕ i=1  n   ∂( f ◦ ϕ−1 ) ∂( f ◦ ϕ−1 ) d(ϕ ◦ γ)i d(ϕ ◦ δ)i = (0) + (0) (ϕ(p)) (ϕ(p)) dt dt ∂ϕi ∂ϕi i=1  n  ∂( f ◦ ϕ−1 ) d(ϕ ◦ γ)i = (0) (ϕ ◦ γ(0)) dt ∂ϕi i=1  ∂( f ◦ ϕ−1 ) d(ϕ ◦ δ)i + (0) (ϕ ◦ δ(0)) dt ∂ϕi n

=

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= ( f ◦ γ)(0) + ( f ◦ δ)(0) = Xγ, p + Xδ, p . Similarly, in order to show that .· : R × Tp M → Tp M is well defined, we must show that there exists a curve .τ such that .λ · Xγ, p = Xτ, p . Define .τ : R → M such that .τ(t) = (γ ◦ μ)(t) and . μ = λt; then .τ(0) = γ(0) = p, and by chain rule, we have .

Xτ, p ( f ) = ( f ◦ γ ◦ μ)(0) = λ( f ◦ γ)(0) = λ · Xγ, p .

Therefore, .Tp M is a vector space; see [Tu] for the proof that the dimension of .Tp M is equal to the dimension of the manifold M. Definition B.5.13 The total space of a tangent bundle of M, denoted by .TM, is defined by TM := {(x, y) : x ∈ M and y ∈ Tx M }  Tp M. =

.

p ∈M

To show that .TM is a manifold of dimension 2n, we refer the reader to [Tu]. The π tangent bundle is .TM −−→ M where the projection .π : TM → M is defined by . Xγ, p → p where p is the point for which . Xγ, p ∈ Tp M. π

Definition B.5.14 A section of the tangent bundle .TM −−→ M is called a vector field on M. More precisely, a vector field is a section .ν : M → TM which assigns to each point . p ∈ M a vector .ν(p) tangent to M at p. In 1966, Emery Thomas coined the term “span" of a manifold in [Thom] while working on stably equivalent vector bundles. Definition B.5.15 Vector fields .ν1, · · · , νk are said to be linearly independent if the k vectors .{ν1 (p), ν2 (p), · · · , νk (p)} are linearly independent for all . p ∈ M. The maximal number of linearly independent vector fields on M is called the span of M and is denoted by .span(M). While not all n-manifolds have a trivial tangent bundle, in the next section, we will show that this is the case for dimension 3; every orientable 3-manifold has a trivial tangent bundle, by using the next theorem. Theorem B.5.16 An n-manifold M has a trivial tangent bundle if and only if span(M) = n.

.

π

Proof Consider the bundle . M × Rn −−−→ M, and let .e1, · · · , en be the basis elements for .Rn ; then .νi : M → M × Rn defined by .νi (p) = (p, ei ) for .1 ≤ i ≤ n are sections,

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and .ν1 (p), · · · , νn (p) are linearly independent for all p. Since the tangent bundle π TM −−→ M is trivial, then there exists a homeomorphism .φ : M × Rn → TM such that .π ◦ φ(p, x ) = p for all . p ∈ M and .x ∈ Rn . Then for .1 ≤ i ≤ n, we have . π ◦ φ ◦ νi (p) = π ◦ φ(p, ei ) = p for all . p ∈ M which implies that . φ ◦ ν1, · · · , φ ◦ νn are sections of the tangent bundle. The n sections .φ ◦ ν1, · · · , φ ◦ νn are linearly independent because .φ| {p }×π −1 ({p }) is a vector space isomorphism for all . p ∈ M which implies that .φ ◦ ν1 (p), · · · , φ ◦ νn (p) are linearly independent for all . p ∈ M. Therefore, .span(M) = n. .

Suppose .span(M) = n; then there exist n linearly independent vector fields {ν1, · · · , νn }, that is, for all . p ∈ M, the n vectors on M, .{ν1 (p), · · · , νn (p)} are n ai νi (p) where .ai ∈ R linearly independent. Let . Xp,γ ∈ Tp M; then . Xp,γ = i=1 n for .1 ≤ i ≤ n. Consider .ϕ : TM → M × R defined by .ϕ(p, νi (p)) = (p, ei (p)) for .1 ≤ i ≤ n where .ei (p) is the basis element for .Rn at the point p. Suppose n n n n . ϕ(p, (p)) = (q, i=1 bi ei (q)) i=1 ai νi (p)) = ϕ(q, i=1 bi νi (q)); then .(p, i=1 ai ei n which implies that . p = q and .ai = bi for all i. For any .(p, a e (p)) ∈ M × Rn , i=1 ii n n n there exists .(p, i=1 ai νi (p)) ∈ TM such that .ϕ(p, i=1 ai νi ) = (p, i=1 ai ei (p)). We n n bi ei (q)) = π(p, i=1 bi νi (p)) = p and .φ restricted to the also have .π ◦ φ−1 (p, i=1 fiber is a vector space isomorphism because it is a map of vector spaces of the same dimension which sends basis elements to basis elements. Therefore, the tangent bundle is trivial.  .

A Whitney sum bundle is created by two vector bundles over the same manifold such that the fiber of the Whitney sum is the direct sum of the fibers of the two bundles. Some authors as in [Hus] loosen the definition by not requiring that the bundles be vector bundles. π1

π2

Definition B.5.17 Let .E1 −−−→ M and .E2 −−−→ M be two vector bundles. We define π . E1 ⊕ E2 − −→ M to be the Whitney sum bundle (direct sum bundle) where .E1 ⊕ E2 = {(v1, v2 ) ∈ E1 × E2 ; π1 (v1 ) = π(v2 )} and .π : (v1, v2 ) → π1 (v1 ) = π2 (v2 ). For an arbitrary . p ∈ M, .π −1 ({p}) = π1−1 ({p}) ⊕ π2−1 ({p}).

B.5.2 Stably Parallelizable and Parallelizable Manifolds π1

Definition B.5.18 Let M be an n-manifold, TM −−−→ M be the tangent bundle π2 over M, and E −−−→ M be a trivial line bundle over M; then M is called stably π parallelizable if the Whitney sum bundle TM ⊕ E −−→ M is a trivial bundle. Examples of stably parallelizable manifolds are S n for all n; this can be shown by utilizing a smooth embedding of S n into Rn+1 . Theorem B.5.19 S n is stably parallelizable for all n.

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Proof Let n ∈ N be arbitrary, S n := {x ∈ Rn+1 ; ||x || = 1}, and consider the tangent π1 bundle of S n , TSn −−−→ S n . Recall in Example B.5.10 we defined the normal bundle π2 of S n into Rn+1 , NSn −−−→ S n , and proved that it is a trivial line bundle. Therefore, in order to prove that S n is stably parallelizable, it suffices to prove that the Whitney π sum bundle TSn ⊕ NSn −−→ S n is trivial. q) ∈ TSn × NSn ; p = π1 (Xγ, p ) = π2 ( q, t q) = q } We have TSn ⊕ NSn = {(Xγ, p, q, t −1 −1 −1 and π ({ p}) = π1 ({ p}) ⊕ π2 ({ p})  Rn ⊕ R. Therefore, the fiber of the Whitney sum bundle is isomorphic to Rn+1 and TSn ⊕ NSn = {(Xγ, p, p, t p) ∈ TSn × NSn ; Xγ, p ∈ Tp M, p ∈ S n, and Xγ, p ⊥ t p}.

.

By considering S n embedded in Rn+1 , we have that for each p ∈ S n , Tp M ⊂ Rn+1 . p, Xγ, p + t p). Define the map ϕ : TSn ⊕ NSn → S n × Rn+1 by (Xγ, p, p, t p) → ( p, Xγ, p + t p) = ( q, Xβ, q + t  q) which Suppose ϕ(Xγ, p, p, t p) = ϕ(Xβ, q, q, t  q); then ( implies that p = q and Xγ, p + t p = Xβ, p + t  p. Since Xγ, p ⊥ t p and Xβ, p ⊥ t  p, then t = t  and Xγ, p = Xβ, p . Let ( p, x ) ∈ S n × Rn+1 ; then x − Proj p x ∈ Tp S n and proj p x = t p for some t ∈ R. p, x ) and π ◦ ϕ−1 ( p, x ) = π(Xp, p, t p) = Let Xp = x − proj p x ; then ϕ(Xγ, p, p, t p) = ( p, t p) = p. π1 (Xp ) = π2 ( For each fixed p ∈ S n , consider the basis for Tp M, {X1, · · · , Xn }; then the set {(X1, p, 0), · · · , (Xn, p, 0), (0, p, t p)} for t ∈ R\{0} forms a basis for TSn ⊕ NSn . For p, Xγ, p + t p), the map restricted to the fiber ϕ| {p }×π −1 ({p }) defined by (Xγ, p, p, t p) → ( p, Xi ) for 1 ≤ i ≤ n and ϕ| {p }×π −1 ({p }) (0, p, t p) = we have ϕ| {p }×π −1 ({p }) (Xi, p, 0) = ( ( p, t p). That is, the image of the linearly independent vectors is linearly independent;  this implies that ϕ| {p }×π −1 ({p }) is a vector space isomorphism. Definition B.5.20 Let W be an n-manifold and M a manifold embedded into W. π Consider the tangent bundle of W, TW −−→ W; then the ambient tangent  bundle over M, denoted by TW | M , is a vector bundle with total space TW | M = p ∈M Tp W, the projection map obtained by restricting π to TW | M , and base space M. The next definition uses the ambient tangent bundle to generalize the definition of the normal bundle that was defined in Example B.5.10 for S n → Rn+1 . Definition B.5.21 Let W be an n-manifold and M be a manifold embedded into W. The normal bundle of M in W is defined as the following quotient bundle: .

N M := TW | M /T M .

Notice that this yields the following bundle isomorphism: T W | M  T M ⊕ N M.

.

(B.2)

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The following proposition was proved by Dale Husemoller in [Hus] by constructing a bundle isomorphism. Proposition B.5.22 ([Hus]) Let W be an (n+1)-dimensional manifold with boundary ∂W  , and let M be an n-manifold such that M = ∂W; then there exists an isomorphism of vector bundles between the ambient tangent bundle over M and the Whitney sum bundle of the tangent bundle of M and a trivial line bundle. That is, TW | M  T M ⊕ (M × R).

.

A nice direct corollary to Proposition B.5.22 arises when the tangent bundle of a manifold with boundary is trivial. Corollary B.5.23 Let W be an n-dimensional manifold with boundary ∂W  . If the tangent bundle of W is trivial, then its boundary is stably parallelizable. Definition B.5.24 M is parallelizable if the tangent bundle of M is trivial. Equivalently, an n-manifold M is parallelizable if span(M) = n. An example of a parallelizable manifold is S 3 ; we will use a well-known technique that utilizes unit quaternions to prove that this is the case. Definition B.5.25 A quaternion is defined by the equation, q = q0 + q1i + q2 j + q3 k, where i 2 = j 2 = k 2 = i j k = −1. It follows that i j = k = − ji, j k = i = −k j, ki = j = −ik.

.

The set of quaternions, denoted by H, forms a noncommutative division ring. The √ ∗  2 magnitude of a quaternion is defined by ||q|| = qq = q0 + q12 + q22 + q32 where q∗ = q0 − q1i − q2 j − q3 k. In scalar-vector notation, q = q0 + v , where v = q1i + q2 j + q3 k; we can define quaternion arithmetic operations in the following way. Let q = q0 + v and qi = wi + vi for i = 1, 2; then we have: • Scalar multiplication: sq = sq0 + sv . • Addition: q1 + q2 = (w1 + w2 ) + (v1 + v2 ). • Multiplication: q1 q2 = (w1 w2 − v1 · v2 ) + (w1v2 + w2v1 + v1 × v2 ), where v1 · v2 is the dot product and v1 × v2 is the cross-product in R3 by associating i, j, andk with the coordinates of R3 . • Conjugation: q∗ = (w, −v ) with the property (q1 q2 )∗ = q∗2 q∗1 . • We define an inner product of H on the basis {1, i, j, k}; for a, b ∈ {1, i, j, k}, a, b = δa,b . In fact, q, q = q02 + q12 + q22 + q32 = qq∗ . While it is well known that the 3-sphere can be described as a subset of R4 , that is, S 3 := {(x1, x2, x3, x4 ) ∈ R4 ; x12 + x22 + x32 + x42 = 1}, another description of the

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3 3-sphere is a subset  of the set of quaternions H containing all unit quaternions S :=

{q ∈ H; ||q|| = q02 + q12 + q22 + q32 = 1}. The elements of unit quaternions can be described as q = cos(α/2) + u sin(α/2); this form is widely using in aerodynamics and robotics because it describes a change in orientation by a rotation of α about an axis pointing in the direction of u. By using this description, we define a curve on S 3 , q(t) : R → S 3 , by t → cos(αt/2) + u sin(αt/2) where ||u || = 1. Then q(t) ∈ TS3 can be simplified to the following: q(t) = −

.

α α sin(tα/2) + u cos(tα/2) 2 2

α (− sin(tα/2) + u cos(tα/2)) 2 α = (−u · u sin(α/2) + u cos(α/2) + u × u sin(α/2)) 2 α = (0 + u)(cos(α/2) + u sin(α/2)) 2 α = uq(t). 2 =

Therefore, for each t ∈ R, q(t) can be decomposed into a linear combination of iq(t), jq(t), kq(t); .

q (t) =

αu2 αu3 αu1 iq(t) + jq(t) + kq(t). 2 2 2

We will use this idea to prove the following theorem by defining three vector fields νi : S 3 → TS3 and proving that for each t, the three quaternions iq(t), jq(t), and kq(t) are linearly independent. Theorem B.5.26 S 3 is parallelizable. π

Proof Let TS3 −−→ S 3 be the tangent bundle of S 3 . By Theorem B.5.16, to prove that S 3 is parallelizable, it suffices to prove that there are three linearly independent vector fields on S 3 , νi : S 3 → TS3 for i = 1, 2, 3. That is, we want to prove that for every point q ∈ S 3 , the vectors ν1 (q), ν2 (q), andν3 (q) are linearly independent. Consider a unit quaternion q = q0 + q1i + q2 j + q3 k, and then define the vector fields as follows: .

ν1 : q → qi = −q1 + q0i − q3 j + q2 k ν2 : q → q j = −q2 − q3 j + q0 j + q2 k ν3 : q → qk = −q3 − q2i + q1 j + q0 k

For two unit quaternions q1 and q2 , we have q1 q2, q1 q2  = q1 q2 (q1 q2 )∗ = q1 q2 q∗2 q∗1 = 1. Let a, b ∈ {1, i, j, k} where a  b, and let q be a unit quaternion; then .

     a+b a+b q √ ,q √ =1 2 2

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and .

     a+b a+b 1 q √ ,q √ = (qa, qa + qa, qb + qb, qa + qb, qb) 2 2 2 1 = 1 + (qa, qb + qb, qa). 2

Therefore, qa, qb = qb, qa = 0, which implies that q, qi, q j, andqk are all orthogonal to each other. Hence, qi, q j, andqk ∈ TS3 are linearly independent, and  ν1, ν2, andν3 are linearly independent vector fields. Theorem B.5.27 ([Kir2]) Let L be an even link; then WL is parallelizable, where WL is a 4-manifold determined by a framed link L , by adding 2-handles to a 4-ball along L . The third author in [Iba] completed Fomenko and Matveev’s proof of Stiefel’s theorem that all orientable 3-manifolds are parallelizable by utilizing results involving stably parallelizable manifolds.8 We will summarize this below. Theorem B.5.28 ([BK]) Let M be an n-dimensional oriented closed stably parallelizable manifold; then span(M) ≥ span(S n ). Theorem B.5.29 ([Sti]) Every orientable 3-manifold is parallelizable. Proof Suppose M is a closed orientable 3-manifolds; then the proof that M is stably parallelizable can be found in [FoMa] and summarized as follows. By Theorem B.4.6, M can be obtained from S 3 by integral surgery along an even link. Furthermore, by Theorem B.5.27, a closed orientable 3-manifold obtained from S 3 by integral surgery along an even link is the boundary of a parallelizable 4-manifold. Both theorems combined implies that M is the boundary of a parallelizable 4-manifold. By Corollary B.5.23, M is stably parallelizable. Now, since S 3 is parallelizable, then span(S 3 ) = 3. By Theorem B.5.28, span(M) = 3; therefore, M is parallelizable. Suppose M is a compact orientable 3-manifold with nontrivial boundary. Let X be the double of M; then M is an embedding of X. Since X is a closed orientable 3-manifold, then it is parallelizable. Recall that the rank of a vector bundle is the dimension of its fiber. Since the normal bundle of M into X, N M, is a 0-ranked vector bundle over M and T X | M  T M ⊕N M, then the tangent bundle of M, T M, is trivial, and hence, M is parallelizable. 

8 See [BaZu, BeLi, DGGK, Gon, Gei, Gon, Kir3, FoMa, Whi5] for various proofs.

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Exercise B.5.30 Show that a closed oriented surface is parallelizable if and only if the Euler characteristic is zero. Hint: For people who know basic differential geometry, use the Gauss-Bonnet theorem.

Appendix C

Table of Knots

This appendix lists Dale Rolfsen’s knots and links table that was compiled by James Bailey and drawn by Ali Roth. The table contains all prime knots of up to ten crossings and links of up to nine crossings excluding trivial and split links. There is a note on knot .10162 pointing out that Kenneth Perko proved that knot .10162 and knot .10161 are equivalent, famously known as Perko’s pair. See Lecture 1 for a discussion on Perko’s pair. In addition to the diagrams, the table provides Conway’s notation and an abbreviation of the Alexander (multivariable) polynomial for knots and links.

Extended A & B notation for the 1 st link of one component and three crossings. Conway’s notation Alexander polynomial .

We refer to Rolfsen’s book [Rol] for details on the notation.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5

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References

[Abe] E. Abe, Hopf algebras, Cambridge Tracts in Mathematics, 74. Cambridge University Press, Cambridge-New York, 1980. xii+284 pp. [Ada] C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, W. H. Freeman and company (1994). (Second edition, American Mathematical Society, 2004.) [Akb] S. Akbulut, On 4-dimensional smooth Poincare conjecture, e-print: arXiv:2209.09968 [math.GT]. [Alb] D. Albers, John Horton Conway−Talking a Good Game, Math. Horizons, vol. 1, no. 2, Spring, 6-9. [Ale1] J. W. Alexander, Letter to Oswald Veblen, 1919, Papers of Oswald Veblen, 1881–1960 (bulk 1920–1960), Archival Manuscript Material (Collection), Library of Congress. [Ale2] J. W. Alexander, A lemma on systems of knotted curves, Proceedings of the National Academy of Sciences of the United States of America. (9) (1923): 23–25. [Ale3] J. W. Alexander, On the deformation of an n-cell, Proceedings of the National Academy of Sciences of the United States of America. (9)12 (1923): 406–407. [Ale4] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928) 275–306. [Ale5] J. W. Alexander, Some Problems in Topology, Verhandlungen Int. Math. Kongr., Zurich, 1932. [AH] P. S. Alexandroff, H. Hopf, Topologie, Springer, Berlin, 1935.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5

465

466

References

[AB] J. W. Alexander, G. B. Briggs, On types of knotted curves, Ann. of Math 28(2), 1926/27, 562–586. [AD] D. André, Solution directe du problème résolu par M. Bertrand, Comp. Rend. Sci. Paris, 105, 1887, 436–437. [AL] C. W. Armond, A. M. Lowrance, Turaev genus and alternating decompositions, Algebr. Geom. Topol., 17 (2017), no. 2, 793–830. [Art1] E. Artin, Theorie der Zöpfe, Abh. Math. Sem. Hamburg, 4, 1925, 47–72. [Art2] E. Artin, Theory of braids, Ann. of Math. (2) 48 (1947), 101–126. [Art3] E. Artin, Braids and permutations, Ann. of Math. (2) 48 (1947), 643–649. [AF] M. M. Asaeda, C. Frohman, A note on the Bar-Natan skein module, Internat. J. Math. 18 (2007), no. 10, 1225–1243. arXiv:math/0602262 [math.QA]. [AP] M. M. Asaeda, J. H. Przytycki, Khovanov homology: torsion and thickness, Advances in topological quantum field theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 179, Kluwer Acad. Publ., Dordrecht, 2004, pp. 135–166. MR 2147419. arXiv:math/0402402. [APS1] M. M. Asaeda, J. H. Przytycki, A. S. Sikora, Categorification of the Kauffman bracket skein module of I-bundles over surfaces, Algebr. Geom. Topol. 4 (2004), 1177–1210. arXiv:math/0409414 [math.QA]. [APS2] M. M. Asaeda, J. H. Przytycki, A. S. Sikora, Kauffman-Harary conjecture holds for Montesinos knots, J. Knot Theory Ramifications 13 (2004), no. 4, 467–477. arXiv:math/0305415 [math.GT]. [Ati1] M. F. Atiyah, The geometry and physics of knots, Miniconference on Geometry and Physics (Canberra, 1989), 1–17, Proc. Centre Math. Anal. Austral. Nat. Univ., 22, Austral. Nat. Univ., Canberra, 1989. [Ati2] M. F. Atiyah, On framings of 3-manifolds, Topology 29 (1990), 1–7. [BaZu] V. Bais, D. Zuddas, On Stiefel’s parallelizability of 3-manifolds, Expo. Math. 41 (2023), no. 1, 238–243. [Bak] R. P. Bakshi, A counterexample to the generalisation of Witten’s conjecture, Contemp. Math. (to appear), e-print: arXiv:2205.01653 [math.GT]. [BGMMP] R. P. Bakshi, H. Guo, G. Montoya-Vega, S. Mukherjee, J. H. Przytycki, The generalized Kauffman-Harary conjecture is true. e-print: arXiv:2301.02645 [math.GT] (submitted). [BIMPW] R. P. Bakshi, D. Ibarra, G. Montoya-Vega, J. H. Przytycki, D. Weeks, On framings of links in 3-manifolds. Canad. Math. Bull. 64 (2021), no. 4, 752–764. arXiv:2001.07782 [math.GT].

References

467

[BIMP1] R. P. Bakshi, D. Ibarra, S. Mukherjee, J. H. Przytycki, A generalization of the Gram determinant of type A. Topology Appl. 295 (2021), Paper No. 107663, 15 pp. arXiv:1905.07834 [math.GT]. [BIMP2] R. P. Bakshi, D. Ibarra, S. Mukherjee, J. H. Przytycki, A Note on the Gram Determinant of Type Mb, Contemporary Mathematics (to appear.) [BLP] R. P. Bakshi, T. T. Q. Lê, J. H. Przytycki, The Kauffman bracket skein module of the connected sum of two solid tori (draft available on request). [BaPr] R. P. Bakshi, J. H. Przytycki, Kauffman bracket skein module of the connected sum of handlebodies: a counterexample, Manuscripta Math. 167 (2022), no. 3-4, 809–820, arXiv:2005.07750 [math.GT]. [BMPSW] R. P. Bakshi, S. Mukherjee, J. H. Przytycki, M. Silvero, X. Wang, On multiplying curves in Kauffman bracket skein algebra of the thickened four-holed sphere. J. Knot Theory Ramifications 30 (2021), no. 14, Paper No. 2141001, 29 pp. arXiv:1805.06062 [math.GT]. [BPW] R. P. Bakshi, J. H. Przytycki, H. Wong, Chapter 63: Skein modules of 3-manifolds, Encyclopedia of Knot Theory, Chapman and Hall/CRC Press (2020), pp. 617–623, ISBN 9781138297845. [BaMe] R. Ball, R. L. Metha, Sequence of invariants for knots and links, J. Physique, 42 (1981), 1193–1199. [Ban] C. Bankwitz, Über die Torsionszahlen der alternierenden Knoten. (German) Math. Ann., 103, 1930, no. 1, 145–161. [BarN1] D. Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Alg. Geom. Top., 2 (2002) 337–370; arXiv:math.QA/0201043. [BarN2] D. Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443–1499. arXiv:math/0410495 [math.GT]. [BarN3] D. Bar-Natan, Fast Khovanov homology computations, J. Knot Theory Ramifications 16 (2007), no. 3, 243–255. arXiv:math/0606318 [math.GT]. [BG] D. Bar-Natan, J. Green, JavaKh – a fast program for computing Khovanov homology, part of the KnotTheory’ Mathematica Package, http://katlas.math.utoronto.ca/wiki/Khovanov_Homology. [Bar] J. W. Barrett, Skein spaces and spin structures, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 2, 267–275. arXiv:gr-qc/9512041. [Bax] R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc., London (1982).

468

References

[BKKPR] The Disc Embedding Theorem, Edited by Stefan Behrens, Boldizsar Kalmar, Min Hoon Kim, Mark Powell, and Arunima Ray, Oxford Univerity Press, 2021. xvii+473 pp. ISBN: 9780198841319. [BeLi] R. Benedetti, P. Lisca, Framing 3-manifolds with bare hands, Enseign. Math. 64 (2018), no. 3-4, 395–413. [BEW] B. C. Berndt, R. J. Evans, Kenneth S. Williams, Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, Volume 21, A Wiley-Interscience Publications, 1998. [Bin1] R. H. Bing, Necessary and sufficient conditions that a 3-manifold be S 3 , Ann. of Math. 68 (1958), 17–37. [Bin2] R. H. Bing, An alternative proof that 3-manifolds can be triangulated, Ann. of Math. (2) 69 (1959), 37–65. [Bin3] R. H. Bing, Correction to “Necessary and sufficient conditions that a 3-manifold be S 3 ,” Ann. of Math. (2) 77 (1963), 210. [Bir1] G. D. Birkhoff, A determinant formula for the number of ways of coloring a map, Ann. of Math. (2)14, 1912, 42–46. [Bir2] G. D. Birkhoff, Fifty years of American Mathematics, Amer. Math. Soc., Semicentennial Addresses, v. 2, 1938, 270–315. Also in George David Birkhoff Collected Mathematical Papers, Vol. 3, AMS (1950), 606–651. [Birm] J. S. Birman, On the Jones polynomial of closed 3-braids, Invent. Math. 81 (1985), no. 2, 287–294. [BC] J. S. Birman, J. Cannon, Braids, Links, and Mapping Class Groups, Princeton University Press (1974). [BW] J. S. Birman, H. Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), no. 1, 249–273. [BHMV1] C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Three-manifold invariants derived from the Kauffman bracket, Topology 31 (1992), no. 4, 685–699. [BHMV2] C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995), no. 4, 883–927. [BlLi] S. A. Bleiler, R. A. Litherland, Lens spaces and Dehn surgery, Proc. Amer. Math. Soc. 107 (1989), no. 4, 1127–1131. [Boe1] O. Boeddicker, Beitrag zur Theorien des Winkels, Dissertation, Göttingen 1876. [Boe2] O. Boeddicker, Erweiterung der Gausschen Theorie der Verschlingungen, Stuttgart, 1876.

References

469

[Bog] M. Bognar, Fundations of linking theory, Akademiai Kiado, Budapest 1992. [BE] V. G. Boltyanskii, V. A. Efremovich, Intuitive combinatorial topology, 1982 (English translation, Springer, 2001). [BO] F. Bonahon, J-P Otal, Scindements de Heegaard des espaces lenticulaires. (French) [Heegaard splittings of lens spaces] Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 3, 451–466 (1984). [BW] F. Bonahon, H. Wong, Quantum traces for representations of surface groups in SL2 (C), Geom. Topol. 15 (2011), no. 3, 1569–1615. arXiv:1003.5250 [math.GT]. [Bou] P. Bousseau, Strong positivity for the skein algebras of the 4punctured sphere and of the 1-punctured torus, Comm. Math. Phys. 398 (2023), no. 1, 1–58. arXiv:2009.02266 [math.GT]. [Bra] H. Brahana, Systems of circuits on two-dimensional manifolds, Ann. of Math. (2) 23 (1921), no. 2, 144–168. [BLM] R. D. Brandt, W. B. R.Lickorish, K. C. Millett, A polynomial invariant for unoriented knots and links, Invent. Math. 84 (1986), 563–573. [Brau] R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. (2) 38 (1937), no. 4, 857–872. [BK] G. E. Bredon, A. Kosinski. Vector Fields on π-Manifolds, Annals of Mathematics, vol. 84, no. 1, 1966, pp. 85–90. [Bre1] E. Breitenberger, Gauss und Listing: Topologie und Freundschaft. (German) [Gauss and Listing: topology and friendship] Gauss-Ges. Gvttingen Mitt. No. 30 (1993), 3–56. [Bre2] E. Breitenberger, Johann Benedikt Listing, History of topology, 909–924, North-Holland, Amsterdam, 1999. (“Indeed, Listing was present when Gauss finally closed his eyes shortly after 1 a.m. on 23 February 1855”.) [Bro] E. J. Brody, The topological classification of the lens spaces, Ann. of Math. (2) 71 (1960), 163–184. [BSST] R. I. Brooks, C. A. B. Smith, A. H. Stone, W. T. Tutte, The dissection of rectangles into squares, Duke Math. J. 7, 1940, 312–340. [Bru] H. Brunn, Topologische Betrachtungen. Zeitschrift fur Mathematik und Physik, 37, 1892, 106–116. [Bul1] D. Bullock, The (2, ∞)-skein module of the complement of a (2, 2p+ 1) torus knot, J. Knot Theory Ramifications 4 (1995), no. 4, 619– 632. [Bul2] D. Bullock, Estimating a skein module with SL2 (C) characters, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1835–1839.

470

References

[Bul3] D. Bullock, Rings of SL2 (C)-characters and the Kauffman bracket skein module, Comment. Math. Helv. 72 (1997), no. 4, 521–542. [Bul4] D. Bullock, On the Kauffman bracket skein module of surgery on a trefoil, Pacific J. Math. 178 (1997), no. 1, 37–51. [Bul5] D. Bullock, A finite set of generators for the Kauffman bracket skein algebra, Math. Z. 231 (1999), no. 1, 91–101. [BLF] D. Bullock, W. Lo Faro, The Kauffman bracket skein module of a twist knot exterior, Algebr. Geom. Topol. 5 (2005), 107–118. arXiv:math/0402102 [math.QA]. [BuPr] D. Bullock, J. H. Przytycki, Multiplicative structure of Kauffman bracket skein module quantizations, Proc. Amer. Math. Soc. 128 (2000), no. 3, 923–931. arXiv:math/9902117 [math.QA]. [BS] I. Burban, O. Schiffmann, On the Hall algebra of an elliptic curve, I. Duke Math. J. 161 (2012), no. 7, 1171–1231. arXiv:math/0505148 [math.AG]. [BZ] G. Burde, H. Zieschang, Knots, De Gruyter, 1985 (second edition 2003). [BuMa] C. Burstin, W. Mayer, Distributive Gruppen von endlicher Ordnung, J. Reine Angew. Math., vol. 160, 1929, pp. 111–130; Translation “Finite Distributive Groups” by Ansgar Wenzel is available at arXiv:1403.6326 [math.GR]. [Bur] B. A. Burton, The Next 350 Million Knots. 36th International Symposium on Computational Geometry, Art. No. 25, 17 pp. LIPIcs. Leibniz Int. Proc. Inform., 164 Schloss Dagstuhl. Leibniz-Zentrum für Informatik, Wadern, 2020. ISBN:978-3-95977-143-6. Classifications DOI:12183. https://regina-normal.github.io/data.html. [CCS] P. Cahn, V. Chernov, R. Sadykov, The number of framings of a knot in a 3-manifold, J. Knot Theory Ramifications 23 (2014), no. 13, 1450072, 9 pp. arXiv:1404.5851 [math.GT]. [Cai] X. Cai, A Gram determinant of Lickorish’s bilinear form, Math. Proc. Cambridge Philos. Soc. 151 (2011), no. 1, 83–94. arXiv:1006.1297v3 [math.GT]. [Cair1] S. S. Cairns, Triangulation of the manifold of class one, Bull. Amer. Math. Soc. 41 (1935), no. 8, 549–552. [Cair2] S. S. Cairns, The manifold smoothing problem, Bull. Amer. Math. Soc. 67 (1961), 237–238. [Car] A. Carrega, Nine generators of the skein space of the 3-torus, Algebr. Geom. Topol. 17 (2017), no. 6, 3449–3460. arXiv:1603.09661 [math.GT].

References

471

[CE] H. Cartan, S. Eilenberg, Homological algebra. With an appendix by David A. Buchsbaum. Reprint of the 1956 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. xvi+390 pp. [CES] J. S. Carter, M. Elhamdadi, and M. Saito, Homology theory for the set-theoretic Yang-Baxter equation and knot invariants from generalizations of quandles, Fund. Math. 184 (2004), 31–54. arXiv:math/0206255 [math.GT]. [CJKLS] J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3947– 3989. arXiv:math/9903135 [math.GT]. [CJKS] S. Carter, D. Jelsovsky, S. Kamada, M. Saito, Quandle homology groups, their Betti numbers, and virtual knots, J. Pure Appl. Algebra 157 (2001), no. 2-3, 135–155. arXiv:math/9909161 [math.GT]. [CKS1] S. Carter, S. Kamada, M. Saito, Geometric interpretation of quandle homology, J. Knot Theory Ramifications, 10, 2001, 345–386. [CKS2] S. Carter, S. Kamada, M. Saito, Surfaces in 4-space, Encyclopedia of Mathematical Sciences, Low-Dimensional Topology III, R. V. Gamkrelidze, V. A. Vassiliev, Eds., Springer-Verlag, 2004, 213pp. [Cau] A. Caudron, Classification des noeuds et des enlacements (French) (Thèse et additifs), Univ. Paris-Sud, 1989. [CL] J. C. Cha, C. Livingston, KnotInfo: Table of Knot Invariants, http://www.indiana.eduknotinfo. [CF] L. O. Chekhov, V. V. Fok, Quantum Teichmüller spaces. (Russian summary) Teoret. Mat. Fiz. 120 (1999), no. 3, 511–528; translation in Theoret. and Math. Phys. 120 (1999), no. 3, 1245–1259. [Che] Q. Chen, The 3-move conjecture for 5-braids, Knots in Hellas’ 98; The Proceedings of the International Conference on Knot Theory and its Ramifications; Volume 1. In the Series on Knots and Everything, Vol. 24, September 2000, pp. 36–47. [ChP1] Q. Chen, J. H. Przytycki, The Gram matrix of the Temperley-Lieb algebra is similar to the matrix of chromatic joins, Commun. Contemp. Math. 10 (2008), 849–855. arXiv:0806.0878 [math.GT]. [ChP2] Q. Chen, J. H. Przytycki, The Gram determinant of the type B Temperley-Lieb algebra, Adv. in Appl. Math., 43(2), 2009, 156– 161. arXiv:0802.1083 [math.GT]. [CMPWY1] Z. Cheng, S. Mukherjee, J. H. Przytycki, X. Wang and S. Y. Yang, Realization of plucking polynomials, J. Knot Theory Ramifications 26( 2017), no. 2, 1740016, 9pp.

472

References

[CMPWY2] Z. Cheng, S. Mukherjee, J. H. Przytycki, X. Wang, S. Y. Yang, Strict unimodality of q-polynomials of rooted trees, J. Knot Theory Ramifications 27 (2018), no. 7, 1841009, 19 pp. arXiv:1601.03465 [math.CO]. [CMPWY3] Z. Cheng, S. Mukherjee, J. H. Przytycki, X. Wang, S. Y. Yang, Rooted trees with the same plucking polynomials, Osaka J. Math. 56 (2019), no. 3, 661–674. arXiv:1702.02004 [math.GT]. [Cher] V. Chernov, Framed knots in 3-manifolds and affine self-linking numbers, J. Knot Theory Ramifications 14 (2005), no. 6, 791–818. arXiv:math/0105139 [math.GT]. [CDM] S. Chmutov, S. Duzhin, J. Mostovoy, Introduction to Vassiliev knot invariants, Cambridge University Press, Cambridge, 2012. xvi+504 pp. [Col] D. Collon, Ancient Near Eastern Art, University of California Press, Berkeley, Los Angeles, 1995. [CGP] F. Costantino, N. Geer, B. Patureau-Mirand, Admissible skein modules. e-print: arXiv:2302.04493 [math.GT]. [Con1] J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) 1970, pp. 329–358 Pergamon, Oxford. [Con2] A. Conway, The Levine-Tristram signature: a survey; 2019–20 MATRIX annals, 31–56, MATRIX Book Ser., 4, Springer, Cham, [2021]. [CLRS] T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to algorithms. Third edition, MIT Press, Cambridge, MA, 2009. xx+1292 pp. [Crow] R. H. Crowell, Nonalternating links, Illinois J. Math. 3 (1959), 101–120. [CF] R. H. Crowell, R. H. Fox, Introduction to knot theory. Reprint of 1963 original, Graduate Texts in Mathematics, No 57, SpringerVerlag, New York-Heidelberg, 1977 ISBN 0-387-90272-4. [CS] M. Culler, P. B. Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. of Math. (2) 117 (1983), no. 1, 109– 146. [Cuo] G. Cuomo, Superfluids, vortices and spinning charged operators in 4d CFT, J. High Energ. Phys. 2020, 119 (2020). https://doi.org/10.1007/JHEP02(2020)119. arXiv:1906.07283v3 . [DJKS] M. K. Da¸bkowski, S. Jablan, N. A. Khan, R. K. Sahi, On 4-move equivalence classes of knots and links of two components, J. Knot Theory Ramifications, 20 (2011), no. 1, 47–90. [daV] L. da Vinci, A. Carlo, Trattato della pittura di Lionardo da Vinci, Soc. Tipogr. de Classici Italiani, 1804. 428 pages. Digitized May 7, 2018, books.google.com.

References

473

[DaLiPr] M. K. Da¸bkowski, C. Li, J. H. Przytycki, Catalan states of lattice crossing, Topology and its Applications, 182, March, 2015, 1–15; e-print: arXiv:1409.4065 [math.GT] [DM] M. K. Da¸bkowski, M. Mroczkowski, KBSM of the product of a disk with two holes and S 1 . Topology Appl. 156 (2009), no. 10, 1831–1849. arXiv:0808.3782 [math.GT]. [DP1] M. K. Da¸bkowski, J. H. Przytycki, Burnside obstructions to the Montesinos-Nakanishi 3-move conjecture. Geom. Topol. 6 (2002), 355–360. arXiv:math/0205040[math.GT]. [DP2] M. K. Da¸bkowski, J. H. Przytycki, Unexpected connection between Burnside groups and Knot Theory, Proc. Nat. Acad. Science, 101(50), December, 2004, 17357–17360. arXiv:math/0309140 [math.GT]. [DP3] M. K. Da¸bkowski, J. H. Przytycki, Catalan states of lattice crossing: an application of the plucking polynomial, Topology and its applications, 254, 12–28, 2019. [DP4] M. K. Da¸bkowski, J. H. Przytycki, Catalan states of lattice crossing: III; in preparation. [DFHHHKPS] S. Das, G. Francis, J. Hart, C. Hartman, J. Heyn-Cubacub, L. H. Kauffman, D. Plepys, D. J. Sandin, Air on the Dirac Strings. https://www.youtube.com/watch. [DFKLS] O. T. Dasbach, D. Futer, E. Kalfagianni, X-S. Lin, N. W. Stoltzfus, The Jones polynomial and graphs on surfaces, J. Comb. Theory, Ser. B, 98, No. 2, 2008, 384–399. [Day] C. L. Day, Quipus and Witches’ knots, With a translation and Analysis of “Oribasus De Laqueis”, The University of Kansas Press, Lawrence 1967. [Deh1] M. Dehn, Jahresbericht der deutschen Mathematiker-Vereinigung, Vol.16, 1907, p.573. [Deh2] M. Dehn, Über die Topologie des dreidimensionalen Raumes. (German) Math. Ann. 69 (1910), no. 1, 137–168. [Deh3] M. Dehn, Lecture notes from Breslau, The Archives of The University of Texas at Austin, 1922. [Deh4] M. Dehn, The mentality of the mathematician. A characterization. Math. Intelligencer 5 (1983), no. 2, 18–26. [Deh5] M. Dehn, Papers on group theory and topology. Translated from the German and with introductions and an appendix by John Stillwell. With an appendix by Otto Schreier. Springer-Verlag, New York, 1987. viii+396 pp. 429

474

References

[DH] M. Dehn, P. Heegaard, Analysis situs, Enzyklopädie der Mathematischen Wissenschaften, vol. III AB3, 153–220, Teubner, Leipzig. [Des] Descartes, R. (1998). Cambridge texts in the history of philosophy. In S. Gaukroger (Ed.), Descartes: The World and Other Writings (Cambridge Texts in the History of Philosophy, p. 209). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511605727.010. [Det] R. Detcherry, Infinite families of hyperbolic 3-manifolds with finite dimensional skein modules. J. Lond. Math. Soc. (2) 103 (2021), no. 4, 1363–1376. arXiv:1903.07686 [math.GT]. [DW] R. Detcherry, M. Wolff, A basis for the Kauffman bracket skein module of the product of a surface and a circle. Algebr. Geom. Topol. 21 (2021), no. 6, 2959–2993. arXiv:2001.05421 [math.GT]. [DL] I. Diamantis, S. Lambropoulou, The braid approach to the HOMFLYPT skein module of the lens spaces L(p, 1). Algebraic modeling of topological and computational structures and applications, 143–176, Springer Proc. Math. Stat., 219, Springer, Cham, 2017. arXiv:1702.06290 [math.GT]. [DiLaPr] I. Diamantis, S. Lambropoulou, J. H. Przytycki, Topological steps toward the Homflypt skein module of the lens spaces L(p, 1) via braids. J. Knot Theory Ramifications 25 (2016), no. 14, 1650084, 26 pp. arXiv:1604.06163 [math.GT]. [Dic] L. E. Dickson, The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Ann. of Math., 11, 1897, pages 65–120, 161–183. [Die] T. Tom Dieck, Knotentheorien und Wurzelsysteme I, II, Math. Gottingensis 21, 1993. [DiF] P. Di Francesco, Meander determinants. Comm. Math. Phys. 191 (1998), no. 3, 543–583. arXiv:hep-th/9612026 [math.GT]. [Don] S. K. Donaldson, An application of gauge theory to fourdimensional topology. J. Differential Geom. 18 (1983), no. 2, 279– 315. [Dun] G. W. Dunnington, Carl Friedrich Gauss, Titan of Science, Hafner Publishing Co., New York, 1955. [DuP] L. G. Du Pasquier, Lèonard Euler et ses amis (Paris: Libraire Scientifique J.Herman, 1929), pp. 131. (English translation by John S.D. Glaus, Leonhard Euler and his friends; Publisher: CreateSpace Independent Publishing Platform (July 18, 2008). [Dür1] A. Dürer, W. Franke, (1923). Albrecht D¨rers Zeichnungen. 20. Aufl. Berlin and Wien: B. Harz.

References

475

[Dür2] A. Dürer, Master Printmaker, Department of Prints and Drawings, Boston Museum of Fine Arts, Hacker Art Books, New York, 1988. [DGGK] S. Durst, H. Geiges, J. Gonzalo Pérez, M. Kegel, Parallelisability of 3-manifolds via surgery. Expo. Math. 38 (2020), no. 1, 131–137. [EZ] S. Eilenberg, J. Zilber, Semi-simplicial complexes and singular homology, Ann. of Math., (2) 51, 1950, 499–513. [Eis] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, 150; Springer-Verlag, New York, 1995. xvi+785 pp. [Eps] D. B. A. Epstein, Curves on 2-manifolds and isotopies. Acta Math. 115 (1966), 83–107. ` d’énumeration, Mém. Acad. Roy. Belgique [Err] A. Errera, Un problme Coll. 8o , (2) 11, 1931, 26 pp. [Eth] M. H. Etherington, Non-associative arithmetics, Proc. Roy. Soc. Edinburgh 62A (1949), 442–453. [Eul] L. Euler, Solutio problematis ad geometriam situs pertinentis, Commentarii Academiae Scientarum Imperialis Petropolitanae 8, 1736, 128–140. [FaMa] B. Farb, D. Margalit, A primer on mapping class groups, Princeton University Press , Princeton, NJ, 2012. xiv+472 pp. ISBN: 978-0691-14794-9 [FR] R. Fenn, C. Rourke, On Kirby Calculus of Links. Topology, 1–15, Pergamon Press Ltd. [FRS] R. Fenn, C. Rourke, B. Sanderson, An introduction to species and the rack space, in M.E.Bozhuyuk (ed), Topics in Knot Theory (Proceedings of the Topology Conference, Erzurum), NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci., 399, Kluver Academic Publishers, 33–35, 1993. [FH] W. Floyd, A. Hatcher, Incompressible surfaces in punctured-torus bundles. Topology Appl. 13 (1982), no. 3, 263–282. [Fox] R. H. Fox, A quick trip through knot theory. Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), 190– 197. [FHLMOY] P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, A. Ocneanu, A new polynomial invariant of knots and links. Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 239–246. [FoGo] V. Fock, A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. No. 103 (2006), 1–211. arXiv:math/0311149 [math.AG].

476

References

[FoMa] A. T. Fomenko, S. V. Matveev, Algorithmic and computer methods for three-manifolds. Translated from the 1991 Russian original by M. Tsaplina and Michiel Hazewinkel and revised by the authors. With a preface by Hazewinkel. Mathematics and its Applications, 425. Kluwer Academic Publishers, Dordrecht, 1997. xii+334 pp. [FP] F. Frankl and L. Pontryagin, Ein Knotensatz mit Anwendung auf die Dimensionstheorie, Math. Ann. 102 (1930), no. 1, 785–789. [Fre] M. H. Freedman, The topology of four-dimensional manifolds. J. Differential Geometry 17 (1982), no. 3, 357–453. [FrKl] R. Fricke, F. Klein, Vorlesungen über die Theorie der automorphen Functionen, Vol. 1, pp. 365–370, B.G. Teubner, Leipzig 1897. Reprint: New York, Johnson reprint Corporation (Academic Press 1965). [FrGe] C. Frohman, R. Gelca, Skein modules and the noncommutative torus, Trans. Amer. Math. Soc. 352, 2000, 4877–4888. arXiv:math/9806107 [math.QA]. [FrKa] C. Frohman, and J. Kania-Bartoszynska, A quantum obstruction to embedding. Math. Proc. Cambridge Philos. Soc. 131 (2001), no. 2, 279–293. [FKL] C. Frohman, J. Kania-Bartoszynska, T. T. Q. Lê, Unicity for representations of the Kauffman bracket skein algebra. Invent. Math. 215 (2019), no. 2, 609–650. arXiv:1707.09234 [math.GT]. [Fus] N. I. Fuss, Solutio quæstionis, quot modis polygonum n laterum in polygonam laterum, per diagonales resolvi quaeat, Nova acta academiæ scientiarum Petropolitanae 9, 1791, 243–251. [GMT] D. Gabai, G. R. Meyerhoff, N. Thurston, Homotopy hyperbolic 3-manifolds are hyperbolic. Ann. of Math. (2) 157 (2003), no. 2, 335–431. arXiv:math/9609207 [math.GT]. [GM1] B. Gabrovšek, M. Mroczkowski, The HOMFLYPT skein module of the lens spaces L p,1 . Topology Appl. 175 (2014), 72–80. [GM2] B. Gabrovšek, M. Mroczkowski, Link diagrams in Seifert manifolds and applications to skein modules. Algebraic modeling of topological and computational structures and applications, 117– 141, Springer Proc. Math. Stat., 219, Springer, Cham, 2017. arXiv:1802.03645 [math.GN]. [Gau1] C. F. Gauss, Disquisitiones arithmeticae. (1801) Translated and with a preface by Arthur A. Clarke. Revised by William C. Waterhouse, Cornelius Greither and A. W. Grootendorst and with a preface by Waterhouse. Springer-Verlag, New York, 1986. xx+472 pp. [Gau2] C. F. Gauss, Summatio quarumdam serierum singularium, Comment. Soc. Reg. Sci. Gottingensis 1 (1811)

References

477

[Gau3] C. F. Gauss, Zur mathematischen Theorie der electrodynamischen Wirkungen, 1833, Werke, Königlichen Gesellschaft der Wissinchaften zu Gottingen, 5, 1877, 602–629. [Gei] H. Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, 109. Cambridge University Press, Cambridge, 2008. xvi+440 pp. [Gi] C. A. Giller, A family of links and the Conway calculus, Trans. Amer. Math. Soc., 270(1), 1982, 75–109. [Gil] P. M. Gilmer, On the Kauffman bracket skein module of the 3-torus. Indiana Univ. Math. J. 67 (2018), no. 3, 993–998. arXiv:1604.02075 [math.GT]. [GH] P. M. Gilmer, J. M. Harris, On the Kauffman bracket skein module of the quaternionic manifold. J. Knot Theory Ramifications 16 (2007), no. 1, 103–125. arXiv:math/0406152 [math.GT]. [GiMa] P. M. Gilmer, G. Masbaum, On the skein module of the product of a surface and a circle. Proc. Amer. Math. Soc. 147 (2019), no. 9, 4091–4106. arXiv:1804.05746 [math.GT]. [GW] P. M. Gilmer, X. Wang, Extra structure and the universal construction for the Witten-Reshetikhin-Turaev TQFT. Proc. Amer. Math. Soc. 142 (2014), no. 8, 2915–2920. arXiv:1201.1921 [math.GT]. [GZ1] P. M. Gilmer, J. K. Zhong, On the Homflypt skein module of S 1 × S 2 . Math. Z. 237 (2001), no. 4, 769–814. arXiv:math/0007125 [math.GT]. [GZ2] P. M. Gilmer, J. K. Zhong, The Homflypt skein module of a connected sum of 3-manifolds. Algebr. Geom. Topol. 1 (2001), 605– 625. arXiv:math/0012056 [math.GT]. [Goe] L. Goeritz, Knoten und quadratische Formen, Math. Z., 36, 1933, 647–654. [Gol] W. M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Invent. Math. 85 (1986), no. 2, 263–302. [Gon] J. Gonzalo, Branched covers and contact structures, Proc. Amer. Math. Soc. 101 (1987), no. 2, 347–352. [GoHa] F. M. Goodman, H. M. Hauschild, Affine Birman-Wenzl-Murakami algebras and tangles in the solid torus. Fund. Math. 190 (2006), 77– 137. arXiv:math/0411155 [math.QA]. [Gor] C. McA. Gordon, Some aspects of classical knot theory, In: Knot theory, L.N.M. 685, 1978, 1–60. [GoLi] C. McA. Gordon, R. A. Litherland, On the signature of a link, Invent. Math., 47(1978), 53–69.

478

References

[GoLu] C. McA. Gordon, J. Luecke, Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. [Gos] T. Gosh, Robion Kirby - History of Low Dimension Topology. Online video clip https://www.youtube.com/watch?v=TkapUOjVV zA&t=158s [GJS] S. Gunningham, D. Jordan, P. Safronov, The finiteness conjecture for skein modules, Invent. Math. 232 (2023), no. 1, 301–363. arXiv:1908.05233 [math.QA]. [Gut] I. Gutman, The matching polynomial, Commun. Math. Comput. Chem. (MATCH), 1979, 75–91. [Hak1] W. Haken, Theorie der Normalflächen. (German) Acta Math. 105 (1961), 245–375. [Hak2] W. Haken, Some results on surfaces in 3-manifolds. 1968 |it Studies in Modern Topology pp. 39–98 Math. Assoc. America, Buffalo, N.Y.; distributed by Prentice-Hall, Englewood Cliffs, N.J. [Ham] A. J. S. Hamilton, The triangulation of 3-manifolds. Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 63–70. [Har] F. Harary, The knots and links of Albrecht Dürer. Atti Accad. Pontaniana (N.S.) 34 (1985), 97–106 (1986). [HaKa] F. Harary, L. H. Kauffman, Knots and graphs. I. Arc graphs and colorings. Adv. in Appl. Math. 22 (1999), no. 3, 312–337. [Har] J. M. Harris, The Kauffman bracket skein module of surgery on a (2,2b) torus link. Pacific J. Math. 245 (2010), no. 1, 119–140. arXiv:0807.1491 [math.GT]. [HJ] P. de la Harpe, V. F. R. Jones, Graph invariants related to statistical mechanical models: Examples and Problems. J. Combin. Theory Ser. B57 (1993), no. 2, 207–227. [HaKaw] R. Hartley, A. Kawauchi, Polynomials of amphicheiral knots, Math. Ann., 243, 1979, 63–70. [Has1] M. G. Haseman, On knots, with a census of the amphicheirals with twelve crossings. Trans. Roy. Soc. Edinburgh, 52 (1917), 235–255 (also Ph.D thesis, Bryn Mawr College, 1918 [Com]). [Has2] M. G. Haseman, Amphicheiral knots. Trans. Roy. Soc. Edinburgh 52 (1919–20), 597–602. [Hash] Y. Hashizume, On the uniqueness of the decomposition of a link, Osaka Math. J., 10, 1958, 283–300. [Hat1] A. E. Hatcher, A proof of the Smale conjecture, Diff(S3)  O(4). Ann. of Math. (2) 117 (1983), no. 3, 553–607.

References

479

[Hat2] A. E. Hatcher, Notes on Basic 3-Manifold Topology, 2007. http://pi.math.cornell.edu/ hatcher/3M/3Mdownloads.html. [Hat3] A. E. Hatcher, Topology of Numbers. (2018 version). Published: American Mathematical Society, Providence, RI, [2022], ©2022. ix+341 pp. ISBN:978-1-4704-5611-5. e-print: http://pi.math.cornell.edu/ hatcher/TN/TNbook.pdf. [HT] A. E. Hatcher, W. P. Thurston, Incompressible surfaces in 2-bridge knot complements. Invent. Math. 79 (1985), no. 2, 225–246. [Hea] M. C. Heath, Early Helladic Clay Sealings from the House of the Tiles at Lerna, Hesperia 27(1958), pl.22, no. S57. [Hee] P. Heegaard, Forstudier til en topologisk Teori for de algebraiske Fladers Sammenhang, Ph.D. thesis, Copenhagen, 1989. [Hem] J. Hempel, 3-Manifolds. Ann. of Math. Studies, No. 86. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. xii+195 pp. [Hir] M. W. Hirsch, Obstruction theories for smoothing manifolds and maps. Bull. Amer. Math. Soc. 69 (1963), 352–356. [Ho] C. F. Ho, A new polynomial for knots and links; preliminary report, Abstracts AMS 6(4) (1985), p 300. [HM] S. Hong, D. McCullough, Mapping class groups of 3-manifolds, then and now. Geometry and topology down under, 53–63, Contemp. Math., 597, Amer. Math. Soc., Providence, RI, 2013. [HW] B. Hopkins, R. J. Wilson, The Truth about Konigsberg, The College Mathematics Journal, May 2004, 198–207. [HKMR] S. Hong, J. Kalliongis, D. McCullough, J. H. Rubinstein,Diffeomorphisms of elliptic 3-manifolds. Lecture Notes in Mathematics, 2055. Springer, Heidelberg, 2012. [HNP] B. Horn, A. Nicolis, and R. Penco, Effective string theory for vortex lines in fluids and superfluids, JHEP 10 (2015) 153 arXiv:1507.05635. [Hor] R. D. Horowitz, Characters of free groups represented in the twodimensional special linear group. Comm. Pure Appl. Math. 25 (1972), 635–649. [Hos] J. Hoste, Chapter 11: Knot Tabulation, Encyclopedia of Knot Theory, Chapman and Hall/CRC Press (2020) pp.77–82, ISBN 9781138297845. [HoKi] J. Hoste, M. Kidwell, Dichromatic link invariants. Trans. Amer. Math. Soc. 321 (1990), no. 1, 197–229. [HP1] J. Hoste, J. H. Przytycki, An invariant of dichromatic links. Proc. Amer. Math. Soc. 105 (1989), no. 4, 1003–1007.

480

References

[HP2] J. Hoste, J. H. Przytycki, Homotopy skein modules of orientable 3-manifolds. Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 3, 475–488. [HP3] J. Hoste, J. H. Przytycki, A survey of skein modules of 3-manifolds. Knots 90 (Osaka, 1990), 363–379, de Gruyter, Berlin, 1992. [HP4] J. Hoste, J. H. Przytycki, The (2, ∞)-skein module of lens spaces; a generalization of the Jones polynomial. J. Knot Theory Ramifications 2 (1993), no. 3, 321–333. [HP5] J. Hoste, J. H. Przytycki, The Kauffman bracket skein module of S 1 × S 2 . Math. Z. 220 (1995), no. 1, 65–73. [HP6] J. Hoste, J. H. Przytycki, The (2, ∞)-skein module of Whitehead manifolds. J. Knot Theory Ramifications 4 (1995), no. 3, 411–427. [HTW] J. Hoste, M. Thistlethwaite, J. Weeks, The First 1701936 Knots, Math. Intell., 20(4), 1998, 33–48. [Hud] J. F. P. Hudson, Piecewise linear topology. University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees W. A. Benjamin, Inc., New York-Amsterdam 1969 ix+282 pp. [Hum] S. P. Humphries. Generators for the mapping class group. Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 44–47, Lecture Notes in Math., 722, Springer, Berlin, 1979. 57N10 (30F35) [Hus] D. Husemoller, Fibre bundles. Third edition. Graduate Texts in Mathematics, 20. Springer-Verlag, New York, 1994. xx+353 pp. [Iba] D. Ibarra, A note on a short proof of the parallelizability of orientable 3-manifolds, eprint: arXiv:2207.12149 [math.GT] (submitted). [IM] D. Ibarra, G. Montoya-Vega, On a new Gram determinant from the Möbius band, eprint: arxiv:2304.05616 [math.GT] (submitted). [Jac] W. H. Jaco, Lectures on three-manifold topology. CBMS Regional Conference Series in Mathematics, 43. American Mathematical Society, Providence, R.I., 1980. xii+251 pp. [JO] W. H. Jaco, U. Oertel, An algorithm to decide if a 3-manifold is a Haken manifold. Topology 23 (1984), no. 2, 195–209. [JS] W. H. Jaco, P. B. Shalen, Seifert fibered spaces in 3-manifolds. Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192 pp. [JVW] F. Jaeger, D. L. Vertigan, D. J. A.Welsh, On the Computational Complexity of the Jones and Tutte Polynomials, Math. Proc. Camb. Phil. Soc. 108, 1990, 35–53. [JaPr] W. Jakobsche, J. H. Przytycki, Topologia 3-wymiarowych rozmaitości, Warsaw University Press, (1987), in Polish (Topology of 3-dimensional manifolds).

References

481

[Jan] A. Januszajtis, Scientist in Old Gdansk: 17th and 18th Centuries (From the history of science in ancient Gdansk), Task Quarterly 5(3), 2001, 289–399. [Jef] L. C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation. Comm. Math. Phys. 147 (1992), no. 3, 563–604. [Joh] K. Johannson, Homotopy equivalences of 3-manifolds with boundaries. Lecture Notes in Mathematics, 761. Springer, Berlin, 1979. ii+303 pp. [Jon1] V. F. R. Jones, Index for subfactors. Invent. Math. 72, 1983, 1–25. [Jon2] V. F. R. Jones, Letter to J.Birman, May 31, 1984. [Jon3] V. F. R. Jones, A polynomial invariant for knots via Non Neumann Algebras. Bull. Amer. Math. Soc. (N.S.) 12 (1985), 103–111. MR 766964 Zbl 0564.57006. [Jon4] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126(2), 1987, 335–388. [Jon5] V. F. R. Jones, On knot invariants related to some statistical mechanical models, Pacific J. Math., 137 (1989), no. 2, 311–334. [Jon6] V. F. R. Jones, Knot theory and statistical mechanics, Scientific American, 263, November, 1990, no. 5, 4, 98–103. [Jon7] V. F. R. Jones, Subfactors and knots, CBMS Regional Conference Series in Mathematics, vol. 80, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1991, MR1134131. [Jon8] V. F. R. Jones, A quotient of the affine Hecke algebra in the Brauer algebra. Enseign. Math. (2) 40 (1994), no. 3–4, 313–344. [Jon9] V. F. R. Jones, Ten problems, in Mathematics: frontiers and perspectives, 79–91, Amer. Math. Soc., Providence, RI, 2000. [JoPr] V. F. R. Jones, J. H. Przytycki, Lissajous knots and billiard knots, Banach Center Publications, Vol. 42, Knot Theory, 1998, 145–163. [Joy] D. E. Joyce. An Algebraic Approach to Symmetry with Applications to Knot Theory. Thesis (Ph.D.)–University of Pennsylvania. 1979. 127 pp. [Kai1] U. Kaiser, Presentations of homotopy skein modules of oriented 3manifolds. J. Knot Theory Ramifications 10 (2001), no. 3, 461–491. arXiv:math/0007031 [math.GT]. [Kai2] U. Kaiser, Quantum deformations of fundamental groups of oriented 3-manifolds. Trans. Amer. Math. Soc. 356 (2004), no. 10, 3869–3880. arXiv:math/0202293 [math.GT].

482

References

[Kai3] U. Kaiser, Frobenius algebras and skein modules of surfaces in 3-manifolds. Algebraic topology—old and new, 59–81, Banach Center Publ., 85, Polish Acad. Sci. Inst. Math., Warsaw, 2009. arXiv:0802.4068 [math.GT]. [Kas] R. M. Kashaev, Quantization of Teichmüller spaces and the quantum dilogarithm. Lett. Math. Phys. 43 (1998), no. 2, 105–115. arXiv:q-alg/9705021[math.QA]. [KT] C. Kassel, V. G. Turaev, Braid groups. With the graphical assistance of Olivier Dodane. Graduate Texts in Mathematics, 247. Springer, New York, 2008. xii+340 pp. [Kau1] L. H. Kauffman, The Conway polynomial, Topology, 20, 1981, 101–108. [Kau2] L. H. Kauffman, A geometric interpretation of the generalized polynomial, preprint 1985. [Kau3] L. H. Kauffman, State models and the Jones polynomial. Topology, 26 (1987), no. 3, 395–407. [Kau4] L. H Kauffman, On Knots, Annals Study No. 115, Princeton University Press (1987). [Kau5] L. H. Kauffman, New invariants in the theory of knots, Amer. Math. Monthly, 95, 1988, 195–242. [Kau6] L. H. Kauffman, An invariant of regular isotopy. Trans. Amer. Math. Soc. 318 (1990), no. 2, 417–471. [Kau7] L. H Kauffman, Knots and Physics. World Scientific Publishers First Edition, Second Edition, Third Edition, Fourth Edition. Series on Knots and Everything 53, World Scientific Publishing Co. Pte. Ltd., Singapore and River Edge and Hackensack, New Jersey 1991, 1994, 2001, 2013. [Kau8] L. H. Kauffman, Formal knot theory, Mathematical Notes 30, Princeton University Press, 1983 (Dover edition 2006). [KL] L. H. Kauffman, S. L. Lins, Temperley-Lieb recoupling theory and invariants of 3-manifolds, Annals of Mathematics Studies, Number 134, Princeton University Press, 1994. [Ker] B. V. Kerékjartô, Vorlessunger über Topologies, I, Flächentopologie, Springer, Berline, 1923. [Kerv] M. A. Kervaire, A manifold which does not admit any differentiable structure. Comment. Math. Helv. 34 (1960), 257–270. [KM] M. A. Kervaire, J. W. Milnor, Groups of homotopy spheres. I. Ann. of Math. (2) 77 (1963), 504–537. [Kho1] M. Khovanov, A categorification of the Jones polynomial. Duke Math. J. 101 (2000), no. 3, 359–426. arXiv:math/9908171 [math.QA].

References

483

[Kho2] M. Khovanov, Personal communication, May 2022. [Kim] S. Kim, Link diagrams with low Turaev genus, Proc. Amer. Math. Soc., 146 (2018), no. 2, 875–890. arXiv:1507.02918 [math.GT]. [Kir1] R. C. Kirby, Problems in low dimensional manifold theory. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif. 1976), Part 2, pp. 273–312, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978. [Kir2] R. C. Kirby, A calculus for framed links in S 3 , Invent. Math. 45 (1978), 36–56. [Kir3] R. C. Kirby, The topology of 4-manifolds. Lecture Notes in Mathematics 1374 (Springer, 1989). [KiMe] R. C. Kirby, P. Melvin, The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2,C). Invent. Math. 105 (1991), no. 3, 473–545 [KiSi] R. C. Kirby, L. C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung. Bull. Amer. Math. Soc. 75 (1969), 742– 749. [Kirc] G. R. Kirchhoff, Über die Auflosung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer strome geführt wird, Annalen d. Physik und Chemie 72 (1847) 497–508. [Kirk] T. P. Kirkman, The enumeration description and construction of knots with fewer than 10 crossing, Trans. R. Soc. Edinburgh, 32, 1885, 281–309. [Kli] M. Kline, Mathematical thought from ancient to modern times. Vol. 1. Second edition. The Clarendon Press, Oxford University Press, New York, 1990. xviii+390+xxii pp. [KnP] M. Kneser, D. Puppe, Quadratische Formen und Verschlin- gungsinvarianten von Knoten, Math. Z., 58, 1953, 376–384. [KAT] The Knot Atlas: K11n34, retrieved August 24, 2022 from: http://katlas.org/wiki/K11n34. [Kno] C. G. Knott, Life and Scientific work of Peter Guthrie Tait, Supplementing the two volumes of scientific papers published in 1898 and 1900, Cambridge University Press, 1911. [KS] K. H. Ko, L. A. Smolinsky, Combinatorial matrix in 3-manifold theory. Pacific J. Math. 149 (1991), no. 2, 319–336. [Kos] A. A. Kosinski, Differential manifolds. Pure and Applied Mathematics, 138. Academic Press, Inc., Boston, MA, 1993. xvi+248 pp.

484

References

[KrMr] P. B. Kronheimer, T. S. Mrowka, Khovanov homology is an unknotdetector. Publ. Math. Inst. Hautes Études Sci. (2011), no. 113, 97– 208.arXiv:1005.4346 [math.GT]. [Kue] H. Kuehnio, Meditationes de quantitatibus imaginariis construendis et radicibus imaginariis exhibendis, Novi Comentarii Academiae Scientiarium Imperalis Petropolitanae, vol III, 1750–1751, 170– 223, 1756. [Kyl] R. H. Kyle, Branched covering spaces and the quadratic forms of links. Ann. of Math. (2)59(1954), 539–548. [LP] S. Lambropoulou, J. H. Przytycki, Hecke algebra approach to skein modules of lens spaces (in preparation). [LR] S. Lambropoulou, C. P. Rourke, Markov’s theorem in 3-manifolds. Special issue on braid groups and related topics (Jerusalem, 1995). Topology Appl. 78 (1997), no. 1–2, 95–122. [Le1] T. T. Q. Lê, The colored Jones polynomial and the A-polynomial of knots. Adv. Math. 207 (2006), no. 2, 782–804. arXiv:math/0407521 [math.GT]. [Le2] T. T. Q. Lê, On Kauffman bracket skein modules at roots of unity. Algebr. Geom. Topol. 15 (2015), no. 2, 1093–1117. arXiv:1312.3705 [math.GT]. [Le3] T. T. Q. Lê, Triangular decomposition of skein algebras. Quantum Topol. 9 (2018), no. 3, 591–632. arXiv:1609.04987 [math.GT]. [Le4] T. T. Q. Lê, Quantum Teichmüller spaces and quantum trace map. J. Inst. Math. Jussieu 18 (2019), no. 2, 249–291. arXiv:1511.06054 [math.GT]. [LTY] T. T. Q. Lê, D. Thurston, T. Yu, Lower and upper bounds for positive bases of skein algebras, Int. Math. Res. Not. IMRN 2021, no. 4, 3186–3202. arXiv:1908.05775. [LT] T. T. Q. Lê, A. Tran, The Kauffman bracket skein module of twobridge links. Proc. Amer. Math. Soc. 142 (2014), no. 3, 1045–1056. arXiv:1111.0332 [math.GT]. [LeTr] T. T. Q. Lê, A. T. Tran, On the AJ conjecture for knots. With an appendix written jointly with Vu Q. Huynh. Indiana Univ. Math. J. 64 (2015), no. 4, 1103–1151. arXiv:1111.5258 [math.GT]. [Leb] V. Lebed, Homologies of algebraic structures via braidings and quantum shuffles. J. Algebra 391 (2013), 152–192. arXiv:1204.3312 [math.CT]. [Lei] G. W. Leibniz, Math. Schriften, 1 Abt., Vol 2, 1850, 19–20. (letter to Christian Huygens (1629–1695), written in 1679).

References

485

[Lev1] J. Levine, The role of the Seifert matrix in knot theory. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 95–98. Gauthier-Villars, Paris, 1971. [Lev2] J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv., 44, 1969, 229–244 [Lic1] W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. of Math., 76, (1962), 531–538. [Lic2] W. B. R. Lickorish, Homeomorphisms of non-orientable twomanifolds, Proc. Camb. Phil. Soc., 59 (1963), pp. 307–317 [Lic3] W. B. R. Lickorish, On the homeomorphisms of a non-orientable surface, Proc. Cambridge Philos. Soc. 61 (1965), 61–64. [Lic4] W. B. R. Lickorish, A relationship between link polynomials, Math. Proc. Cambridge Phil. Soc., 100, 1986, 109–112. [Lic5] W. B. R. Lickorish, Three-manifolds and the Temperley-Lieb algebra, Math. Ann. 290, 1990, 657–670. [Lic6] W. B. R. Lickorish, Invariants for 3-manifolds from the combinatorics of the Jones polynomial, Pacific Journ. Math.,149(2), 1991, 337–347. [Lic7] W. B. R. Lickorish, Calculations with the Temperley-Lieb algebra, Comm. Math. Helv., 67, 1992, 571–591. [Lic8] W. B. R. Lickorish, An introduction to knot theory. (English summary) Graduate Texts in Mathematics, 175. Springer-Verlag, New York, 1997. [LiMi] W. B. R. Lickorish, K. C. Millett, An evaluation of the F-polynomial of a link. Differential topology (Siegen, 1987), 104–108, Lecture Notes in Math., 1350, Springer, Berlin, 1988. [LT] W. B. R.Lickorish, M. B. Thistlethwaite, Some links with nontrivial polynomials and their crossing-numbers, Comment. Math. Helv., 63, 1988, 527–539. [Lie] J. Lieberum, Skein modules of links in cylinders over surfaces. Int. J. Math. Math. Sci. 32 (2002), no. 9, 515–554. arXiv:math/9911174 [math.QA]. [LMT] R. Lidl, G. L. Mullen,G. Turnwald, Dickson polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, 65. Longman Scientific and Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. vi+207 pp. [Lis] J. B. Listing, Vorstudien zur Topologie, Gottinger Studien (Abtheilung 1) 1, 1847, 811–875.

486

References

[Lit1] C. N. Little, On knots, with a census for order 10, Trans. Connecticut Acad. Sci., 18, 1885, 374–378 (also Ph.D. thesis, 1885, Yale University [Com]). [Lit2] C. N. Little, Non-alternate knots, Trans. Royal. Soc. Edinburgh, XXXIX (part III (No.30) – for sessions 1898–99), 1900, 771–778. (Communicated by Professor Tait. (With three plates). (Read July 3, 1899). [Lod] J-L. Loday, Cyclic Homology, Grund. Math. Wissen. Band 301, Springer-Verlag, Berlin, 1992 (second edition, 1998). [MaCu] E. MacCurdy, The Notebooks of Leonardo, New York: Reynal and Hitchcock, 1938, vol.2, p.588. [Mag] W. Magnus, M. Dehn, Math. Intelligencer 1 (1978), 132–143. (Also in: Wilhelm Magnus Collected Paper, Edited by G.Baumslag and B.Chandler, Springer-Verlag, 1984). [Man] C. Manolescu, Pin(2)-equivariant Seiberg-Witten Floer homology and the triangulation conjecture. J. Amer. Math. Soc. 29 (2016), no. 1, 147–176. arXiv:1303.235 [math.GT]. [Mar] A. Markov, Über die freie Äquivqlenz geschlossener Zöpfe, Recueil Math., Moscou 1 (1935) [Mart] B. Martelli, An introduction to geometric topology, CreateSpace Independent Publishing Platform. October 2016. arXiv:1610.02592. [MaSa1] P. Martin, H. Saleur, On an algebraic approach to higher dimensional statistical mechanics. Commun. Math. Phys., 158, 1993, 155–190. [MaSa2] P. Martin, H. Saleur, The blob algebra and the periodic TemperleyLieb algebra. Lett. Math. Phys., 30, 1994, no. 3, 189–206. [MH] J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, 2003, xiii+ 341 pages. [Mas] W. S. Massey, Algebraic Topology: An Introduction, Harcourt, Brace & World, Inc. 1967 (Corrected Eight Printing, SpringerVerlag, 1989). [Mat] G. Mathews (1917). Combinatory Analysis. Vol. II. By Major P. A. Macmahon. Pp. xix 340. 18s. net. 1916. (Camb. Univ. Press.). The Mathematical Gazette, 9(128), 52–52. https://doi.org/10.2307/ 3605230 [MaSo] T. W. Mattman, P. Solis, A proof of the Kauffman-Harary conjecture. Algebr. Geom. Topol. 9 (2009), no. 4, 2027–2039. arXiv:0906.1612 [math.GT]. [Max] J. C. Maxwell, A treatise on electricity and magnetism, Oxford 2 (1873).

References

487

[McC] J. McCool, Some finitely presented subgroups of the automorphism group of a free group. J. Algebra 35, 205–213 [McCu1] D. McCullough, Mappings of reducible 3-manifolds. Geometric and algebraic topology, 61–76, Banach Center Publ., 18, PWN, Warsaw, 1986. [McCu2] D. McCullough, Homeomorphisms which are Dehn twists on the boundary. Algebr. Geom. Topol. 6 (2006), 1331–1340. arXiv:math/0604358 [math.GT]. [MM] D. McCullough, A. Miller, Homeomorphisms of 3-manifolds with compressible boundary. Mem. Amer. Math. Soc. 61 (1986), no. 344, xii+100 pp. [McM] D. R. McMillan Jr., Some contractible open 3-manifolds. Trans. Amer. Math. Soc. 102 (1962), 373–382. [MeTh] W. Menasco, M. Thistlethwaite. The Classification of Alternating Links. Annals of Mathematics. 138 (1: 113–171). https:// www.jstor.org/stable/2946636?origin=crossref. https://doi.org/10. 2307/2946636. [Met] Metropolitan Museum of Art. 2000. “Departmental Accessions: Ancient Near East.” Annual Report of the Trustees of the Metropolitan Museum of Art, No. 130 (Jul. 1,1999–Jun. 30, 2000), p. 9. [Mil1] J. Milnor, On manifolds homeomorphic to the 7-sphere. Ann. of Math. (2) 64 (1956), 399–405. [Mil2] J. Milnor, Differentiable manifolds which are homotopy spheres, Princeton, 1959. Unpublished lecture notes. [Mil3] J. Milnor, A unique decomposition theorem for 3-manifolds. Amer. J. Math. 84 (1962), 1–7. [Mil4] J. Milnor, Morse theory. Based on lecture notes by M. Spivak and R. Wells, Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963 vi+153 pp. [MiSt] J. Milnor, J. D. Stasheff, Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. vii+331 pp. [Mob1] A. F. Möbius. Zur theorie der polyëder und der elementarverwandtschaft. Oeuvres Complètes, Tome 2: 519–559, 1861. [Mob2] A. F. Möbius, Theorie der elementaren Verwandtschaft. Leipziger Sitzungsberichte math.-phys. Classe 15 (1869), also in Werke Bd 2. [Mof] Moffatt K. (2008) Vortex Dynamics: The Legacy of Helmholtz and Kelvin. In: Borisov A.V., Kozlov V.V., Mamaev I.S., Sokolovskiy M.A. (eds) IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence. IUTAM Bookseries, vol 6. Springer, Dordrecht.

488

References

[Moi] E. E. Moise, Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung. Ann. of Math. (2) 56 (1952), 96–114. [Mor1] H. Morton, Infinitely many fibered knots having the same Alexander polynomial, Topology 17 (1978), 101–104. [Mor2] H. R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Phil. Soc. 99, 1986, 107–109. [Mor3] H. R. Morton, Mutant knots, In: New ideas in low dimensional topology, 379–412, Ser. Knots Everything, 56, World Sci. Publ., Hackensack, NJ, 2015. [MPS] H. R. Morton, A. Pokorny, P. Samuelson, The Kauffman skein algebra of the torus. Int. Math. Res. Not. IMRN (2023), no. 1, 855– 900. arXiv:1909.11208 [math.QA]. [MoSa] H. R. Morton, P. Samuelson, The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra. Duke Math. J. 166 (2017), no. 5, 801–854. arXiv:1410.0859 [math.QA]. [MT] H. R. Morton, P. Traczyk, Knots and algebras, Contribuciones Matematicas en homenaje al Profesor D. Antonio Plans Sanz de Bremond, ed. E. Martin-Peinador and Rodez Usan, University of Zaragoza (1990), 201–220. [Mos] L. Moser, Elementary surgery along a torus knot, Pacific J. Math., 38 (1972), 737–745. [Mro1] M. Mroczkowski, Polynomial invariants of links in the projective space. Fund. Math. 184 (2004), 223–267. arXiv:math/0312205 [math.GT]. [Mro2] M. Mroczkowski, Kauffman bracket skein module of a family of prism manifolds. J. Knot Theory Ramifications 20 (2011), no. 1, 159–170. [Mro3] M. Mroczkowski, Kauffman bracket skein module of the connected sum of two projective spaces. J. Knot Theory Ramifications 20 (2011), no. 5, 651–675. arXiv:1008.1007 [math.GT]. [Mro4] M. Mroczkowski, The Dubrovnik and Kauffman skein modules of the lens spaces L p,1 , J. Knot Theory Ramifications, 27, 2018, no. 3, 1840004, 15 pp. [Muk] S. Mukherjee, On odd torsion in even Khovanov homology. Exp. Math. 31 (2022), no. 3, 1039–1045. arXiv:math/1906.06278 [math.GT]. [MPSWY] S. Mukherjee, J. H. Przytycki, M. Silvero, X. Wang, and S. Y. Yang, Search for torsion in Khovanov homology. Exp. Math. 27 (2018), no. 4, 488–497. arXiv:1701.04924v1 [math.GT].

References

489

[MuSc] S. Mukherjee, D. Schütz, Arbitrarily large torsion in Khovanov cohomology. Quantum Topology. (accepted) arXiV:1909.07269 [Mul] G. L. Mullen, Dickson Polynomial, in M. Hazewinkel Encyclopaedia of Mathematics, available at the Springer link: http://encyclopediaofmath.org/index.php?title=Dickson_polynomial &oldid=49965. [Mull] G. Muller, Skein and cluster algebras of marked surfaces. Quantum Topol. 7 (2016), no. 3, 435–503. arXiv:1204.0020 [math.QA]. [Mur] J. Murakami, The Kauffman polynomial of links and representation theory. Osaka J. Math. 24 (1987), no. 4, 745–758. [Mura1] K. Murasugi, On a certain numerical invariant of link types, Trans. A.M.S., 117,1965, p. 387–422. [Mura2] K. Murasugi, Jones polynomial and classical conjectures in knot theory. Topology 26 (1987), no. 2, 187–194. [Mura3] K. Murasugi, Jones polynomial and classical conjectures in knot theory, II, Math. Proc. Camb. Phil. Soc., 102, 1987, 317–318. [Mura4] K. Murasugi, Jones polynomials of periodic links, Pac. J. Math., 131, 1988, 319–329. [Murd] D. C. Murdoch, Quasi-groups which satisfy certain generalized laws, American Journal of Math., 16, 1939, pp. 509–522. [Neu] L. P. Neuwirth, Knots, Groups, and 3-manifolds, papers dedicated to the memory of R. H. Fox), Ann. of Math. Studies, No. 84, Princeton University Press, Princeton, N.J., 1975. [NiPr] M. Niebrzydowski, J. H. Przytycki, Entropic magmas, their homology, and related invariants of links and graphs. Algebr. Geom. Topol. 13 (2013), no. 6, 3223–3243. arXiv:math/1211.2951[math.GT]. ´ [Nie] J. Nielsen, Kurvennetze auf Fl’achen. Inaugural-Dissertation, Kiel 1913, 58 pp. [Ozh] E. P. Ozhigova, On the correspondence of Daniel Bernoulli with N. I. Fuss. Voprosy Istor. Estestvoznan. i Tekhn., 1, 1981, 108–115. [ORS] P. S. Ozsvath, J. Rasmussen, and Z. Szabo. Odd Khovanov Homology. Algebraic Geom. Topol. 13:3 (2013), 1465–1488. e-print, arXiv:math/0710.4300 [math.QA]. [OS] P. Ozsvath, Z. Szabo, On the Heegaard Floer homology of branched double covers, Adv. Math., 194, 2005, 1–33. [PPS] M. D. Pabiniak, J. H. Przytycki, R. Sazdanovć, On the first group of the chromatic cohomology of graphs. Geom. Dedicata 140 (2009), 19–48. arXiv:math/0607326 [math.GT].

490

References

[Pap1] Ch. Papakyriakopoulos, A new proof of the invariance of the homology groups of a complex. Bull. Soc. Math. Grèce 22, 1–154 (1946). [Pap2] C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots. Ann. of Math. (2) 66 (1957), 1–26. [PaSa] V. Pasquier, H. Saleur, Common structures between finite systems and conformal field theories through quantum groups Nucl. Phys. B330, 1990 523–556. [PH] R. C. Penner, J. L. Harer, Combinatorics of train tracks. Annals of Mathematical Studies, 125, Princeton University Press, Princeton, NJ, 1992. xii+216 pp. ISBN: 0-691-08764-4; 0-691-02531-2. [Per1] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, e-print: arXiv:math/0211159 [math.DG]. [Per2] G. Perelman, Ricci flow with surgery on three-manifolds, e-print: arXiv:math/0303109 [math.DG]. [Per3] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, e-print: arXiv:math/0307245 [math.DG]. [Perk1] K. A. Perko Jr. On the classification of knots, Proc. Amer. Math. Soc., 45 (1974), 262–266. [Perk2] K. A. Perko Jr., NAMS 25(6), 1978. [Perk3] K. A. Perko, Invariants of 11-crossing knots, Publications Math. d’Orsay, 1980. [Perk4] K. A. Perko, “Remarks on the history of the classification of knots". Banach Center Publications 103.1 (2014): 241–250. http://eudml.org/doc/282326. [Perk5] K. A. Perko. Historical highlights of non-cyclic knot theory, Journal of Knot Theory and Its Ramifications 25 (2016), no. 03, 1640010. [Poi1] H. Poincare, Analysis Situs (12), Journal d’Ecole Polytechnique Normale, 1 (1895), 1–121. [Poi2] H. Poincare, Cinquième complément à l’Analysis Situs, Rend. Circ. Mat. Palermo 18 (1904), 45–110. [Pot] H. S. A. Potter, On the latent roots of quasi-commutative matrices. Amer. Math. Monthly, 57, 1950, 321–322 [Prz1] J. H. Przytycki, Survey on recent invariants in classical knot theory, Warsaw University, Preprints 6,8,9; Warszawa, 1986; e-print: http://front.math.ucdavis.edu/0810.4191 [Prz2] J. H. Przytycki, tk -moves on links, In Braids, ed. J.S.Birman and A.Libgober, Contemporary Math. Vol. 78, 1988, 615–656; e-print: arXiv: 0606633 [math.GT]

References

491

[Prz3] J. H. Przytycki, On Murasugi’s and Traczyk’s criteria for periodic links. Math Ann., 283, 1989, 465–478. [Prz4] J. H. Przytycki, Skein modules of 3-manifolds. Bull. Polish Acad. Sci. Math. 39 (1991), no. 1–2, 91–100. arXiv:math/0611797 [math.GT]. [Prz5] J. H. Przytycki, Skein module of links in a handlebody, Topology ’90 (Columbus, OH, 1990), 315–342, Ohio State Univ. Math. Res. Inst. Publ., 1, de Gruyter, Berlin, 1992. [Prz6] J. H. Przytycki, Quantum group of links in a handlebody Contemporary Math: Deformation Theory and Quantum Groups with Applications to Mathematical Physics, M. Gerstenhaber and J. D. Stasheff, Editors, Volume 134, 1992, 235–245. [Prz7] J. H. Przytycki, Vassiliev-Gusarov skein modules of 3-manifolds and criteria for periodicity of knots, Low-Dimensional Topology, Knoxville, 1992 ed.: Klaus Johannson, International Press Co., Cambridge, MA 02238, 1994, 157–176. [Prz8] J. H. Przytycki, Teoria we¸złów: podejście kombinatoryczne, (Knots: combinatorial approach to the knot theory), Script, Warsaw, August 1995, 240+XLVIIIpp. [Prz9] J. H. Przytycki, A simple proof of the Traczyk-Yokota criteria for periodic knots, Proc. Amer. Math. Soc., 123, 1995, 1607–1611. [Prz10] J. H. Przytycki, Algebraic topology based on knots: an introduction, Knots 96, Proceedings of the Fifth International Research Institute of MSJ, edited by Shin’ichi Suzuki, 1997 World Scientific Publishing Co., 279–297. [Prz11] J. H. Przytycki, Classical roots of Knot Theory, Chaos, Solitons and Fractals, Vol. 9 (No. 4-5), 1998, 531–545. [Prz12] J. H. Przytycki, A q-analogue of the first homology group of a 3manifold, Contemporary Mathematics 214, Perspectives on Quantization (Proceedings of the joint AMS-IMS-SIAM conference on Quantization, Mount Holyoke College, 1996); Ed. L.A.Coburn, M.A.Rieffel, AMS 1998, 135–144. [Prz13] J. H. Przytycki, 3-coloring and other elementary invariants of knots, Banach Center Publications, Vol. 42, Knot Theory, 1998, 275–295. arXiv:0608172 [math.GT] [Prz14] J. H. Przytycki, Homotopy and q-homotopy skein modules of 3manifolds: an example in algebra situs. Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman (New York, 1998), 143–170, AMS/IP Stud. Adv. Math., 24, Amer. Math. Soc., Providence, RI, 2001. arXiv:math/0402304 [math.GT].

492

References

[Prz15] J. H. Przytycki, Fundamentals of Kauffman bracket skein modules, Kobe Math. J., 16(1), 1999, 45–66. arXiv:math/9809113 [math.GT]. [Prz16] J. H. Przytycki, The Kauffman Bracket skein algebra of a surface times the interval has no zero divisors, Proceedings of the 46th Japan Topology Symposium at Hokkaido University (July 26–29, 1999), 52–61. [Prz17] J. H. Przytycki, Symmetric knots and billiard knots. Ideal knots, 374–414. Ser. Knots Everything, 19 World Scientific Publishing Co., Inc., River Edge, NJ, 1998. arXiv:math/0405151 [math.GT]. [Prz18] J. H. Przytycki, Kauffman bracket skein module of a connected sum of 3-manifolds. Manuscripta Math. 101 (2000), no. 2, 199–207. arXiv:math/9911120 [math.GT]. [Prz19] J. H. Przytycki, The interrelation of the Development of Mathematical Topology in Japan, Poland and USA: Notes to the early history of the Knot theory in Japan, Annals of the Institute for Comparative Studies of Culture, TWCU, Vol. 63, 2002, 61–86. arXiv:math/0108072v1 [math.HO]. [Prz20] J. H. Przytycki, History of knot theory, 2007, e-print: arXiv:math/0703096v1 [math.HO]. [Prz21] J.H. Przytycki, From Goeritz matrices to quasi-alternating links. The mathematics of knots, 257–316. Contrib. Math. Comput. Sci., 1 Springer, Heidelberg, 2011 arXiv:0909.1118 [math.GT]. [Prz22] J. H. Przytycki, When the theories meet: Khovanov homology as Hochschild homology of links, Quantum Topology, 1, no. 2, 93– 109, 2010, arXiv:math/0509334v2 [math.QA]. [Prz23] J. H. Przytycki, Distributivity versus associativity in the homology theory of algebraic structures, Demonstratio Math. 44 (2011), no. 4, 823–869. arXiv:1109.4850 [math.GT]. [Prz24] J. H. Przytycki, The Trieste look at Knot Theory, the chapter in the book Introductory Lectures on Knot Theory: Selected Lectures presented at the Advanced School and Conference on Knot Theory and its Applications to Physics and Biology, ICTP, Trieste, Italy, 11–29 May 2009, World Scientific, Series on Knots and Everything – Vol. 46, 2011/2012, 407–441. arXiv:math/1105.2238 [math.GT]. [Prz25] J. H. Przytycki, Knot Theory and related with knots distributive structures; Thirteen Gdansk Lectures, Gdansk University Press, in Polish, June, 2012, pp. 115. (Second, extended edition: twelve new Gdansk Lectures added, February 2016, pp. 188.). ISBN 978-837865-330-1.

References

493

[Prz26] J. H. Przytycki, Knots and distributive homology: from arc colorings to Yang-Baxter homology, Chapter in: New Ideas in Low Dimensional Topology, World Scientific, Vol. 56, March-April 2015, 413–488. e-print: arXiv:math/1409.7044 [math.GT]. [Prz27] J. H. Przytycki, On Slavik Jablan’s work on 4-moves. J. Knot Theory Ramifications 25 (2016), no. 9, 1641014, 26 pp. arXiv:1512.09162 [math.GT]. [Prz28] J. H. Przytycki, q-polynomial invariant of rooted trees; Arnold Mathematical Journal, 2(4), 449–461, 2016; E-print: arXiv:math/1512.03080 [math.CO]. [Prz29] J. H. Przytycki, Knots and Graphs: two centuries of interaction; in Knot Theory and Its Applications Contemporary Mathematics, Vol. 670, 171–257, September, 2016. [Prz30] J. H. Przytycki, Nonorientable, incompressible surfaces in punctured-torus bundles over S1. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3, 1975–2000. [Prz31] J. H. Przytycki, Knots: From combinatorics of knot diagrams to the combinatorial topology based on knots (chapter IX). Cambridge University Press (to appear), 2025, 950 pp. e-print: arXiv:math/0602264 [math.GT]. [Prz32] J. H. Przytycki, Personal recollection of Vaughan F. R. Jones (1952– 2020), Journal of Knot Theory and Its Ramifications, to appear, 2023. [PrSa] J. H. Przytycki, R. Sazdanović, Torsion in Khovanov homology of semi-adequate links, Fundamenta Mathematicae, 225, 2014, 277– 303, e-print: arXiv:math/1210.5254 [math.QA]. [PS1] J. H. Przytycki, A. S. Sikora, Polygon dissections and Euler, Fuss, Kirkman and Cayley numbers, Journal of Combinatorial Theory series A, 92, 2000, 68–76; e-print: arXiv: 9811086 [math.GT] [PS2] J. H. Przytycki, A. S. Sikora, SUn -quantum invariants for periodic links, the AMS volume on Diagrammatic Morphisms and Applications, Contemporary Mathematics, 318, 2003, 199–205. [PS3] J. H. Przytycki, A. S. Sikora, On skein algebras and SL(2, C)character varieties. Topology 39 (2000), no. 1, 115–148. arXiv:q-alg/9705011[math.QA]. [PS4] J. H. Przytycki, A. S. Sikora, Skein algebras of surfaces. Trans. Amer. Math. Soc. 371 (2019), no. 2, 1309–1332. arXiv:1602.07402[math.QA]. [PrSi] J. H. Przytycki, M. Silvero, Khovanov homology, wedges of spheres and complexity. e-print: arXiv:2305.18648 [math.GT].

494

References

[PT1] J. H. Przytycki, P. Traczyk, Invariants of links of Conway type. Kobe J. Math. 4 (1987), no. 2, 115–139. [PT2] J. H. Przytycki, P. Traczyk, Conway algebras and skein equivalence of links, Proc. Amer. Math. Soc., 100(4), 1987, 744–748. [PW1] J. H. Przytycki, X. Wang, Equivalence of two definitions of set-theoretic Yang-Baxter homology, Journal of Knot Theory and Its Ramifications, 27(7), June 2018, 1841013 (15 pages). arXiv:1611.01178 [math.GT]. [PW2] J. H. Przytycki, X. Wang, The second Yang-Baxter homology for the Homflypt polynomial. e-print: arXiv:2004.07413v1 [math.GT]. [PY] J. H. Przytycki, A. Yasukhara, Symmetry of links and classification of lens spaces. Geom. Dedicata 98 (2003), 57–61. arXiv:math/0011119 [math.GT]. [PZ] J. H. Przytycki, X. Zhu, Gram determinant of planar curves. Involve, Vol. 3, No. 2, 2010, 149–169. arXiv:0810.4649v1 [math.GT]. [Pur] J. Purcell, Hyperbolic knot theory. Graduate Studies in Mathematics, 209. American Mathematical Society, Providence, RI, 2020. xviii+369 pp. [Put] K. K. Putyra, A 2-category of chronological cobordisms and odd Khovanov homology. Knots in Poland III. Part III, 291–355. Banach Center Publ., 103 Polish Academy of Sciences, Institute of Mathematics, Warsaw, 2014. arXiv:math/1310.1895. [Que] H. Queffelec, gl 2 foam functoriality and skein positivity. e-print: arXiv:2209.08794 [math.QA]. [Rad] T. Radó. Über den Begriff der Riemannschen Fläche. Acta Sci. Math. (Szeged) 2 (1925), 101121. [Rae] J. Raeder, Oribassi Collectionum Medicarum Reliquiae, 4 volumes, Leipzig and Berlin, 1928–1933. [Rei1] K. Reidemeister, Elementare Begründung der Knotentheorie, Abh. Math. Sem. Univ. Hamburg, 5 (1), 1927, 24–32. [Rei2] K. Reidemeister, Knotentheorie. Ergebn. Math. Grenzgeb., Bd.1; Berlin: Springer-Verlag (1932). Translated from the German by Leo F. Boron, Charles O. Christenson and Bryan A. Smith. BCS Associates, Moscow, Idaho, 1983. xv+143 pp. [Rei3] K. Reidemeister, Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg 9 (1933), 189–194. [Rei4] K Reidemeister, Homotopieringe und Linsenräume. (German) Abh. Math. Sem. Univ. Hamburg 11 (1935), no. 1, 102–109. [RT] N. Yu. Reshetikhin and V. G Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597.

References

495

[Rio] J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle. Math. Comp. 29 (1975), 215–222. [Rob1] J. D. Roberts, Quantum invariants via skein theory, Phd. Thesis, University of Cambridge (April 1994). [Rob2] J. D. Roberts, Kirby calculus in manifolds with boundary. Turkish J. Math. 21 (1997), no. 1, 111–117. [Rol] D. Rolfsen, Knots and links, Publish or Perish, 1976 (second edition, 1990; third edition, AMS Chelsea Publishing, 2003). [RoSm] A. B. Romanowska and J. D. H. Smith, Modes, World Scientific, 2002, pp. xi+623. [RoSa] C. P. Rourke, B. J. Sanderson, Introduction to piecewise-linear topology. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69. Springer-Verlag, New York-Heidelberg, 1972. viii+123 pp. [Rud] Y. Rudyak, Piecewise linear structures on topological manifolds. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. xxii+106 pp. arXiv:math/0105047 [math.AT]. [Rus] H. Russell, The Bar-Natan skein module of the solid torus and the homology of (n, n) Springer varieties. Geom. Dedicata 142 (2009), 71–89. arXiv:0805.0286 [math.GT]. [Sav] N. Saveliev, Lectures on the topology of 3-manifolds. An introduction to the Casson invariant. Second revised edition. De Gruyter Textbook. Walter de Gruyter & Co., Berlin, 2012. [Sch] F. Schmidt, Problems related to type-A and type-B matrices of chromatic joins. Special issue on the Tutte polynomial, Adv. in Appl. Math. 32 (2004), no. 1–2, 380–390. [Schu1] H. Schubert, Die eindeutige Zerlegbarkeit eines Knoten in Primknoten, Sitzungsber. Acad. Wiss. Heildernerg, Math.- nat. Kl., 3, 1949, 57–104. [Schu2] H. Schubert, Knoten und Vollringe. (German) Acta Math. 90 (1953), 131–286. [Schu3] H. Schubert, 1961: Bestimmung der Primfaktorzerlegung von Verkettungen. Math. Z., 76 (1961), 116–148 [Schü] D. Schütz, (2020). Torsion calculations in Khovanov cohomology. Journal of Knot Theory and Its Ramifications, 29 (2020), no. 8, 2071001, 7 pp. https://doi.org/10.1142/S0218216520710017. [Schüt] M. P. Schützenberger, Une interprétation de certaines solutions de l’Équation fonctionnelle: F(x + y) = F(x)F(y), C. R. Acad. Sci. Paris, 236 (1953) 352–353. [Sei] H. Seifert, Uber das Gaschlecht von Knoten, Math. Ann.110 (1934), 571–592.

496

References

[ST] H. Seifert, W. Threlfall, Seifert and Threlfall: a textbook of topology. Translated from the German edition of 1934 by Michael A. Goldman. With a preface by Joan S. Birman. With "Topology of 3dimensional fibered spaces” by Seifert. Translated from the German by Wolfgang Heil. Pure and Applied Mathematics, 89. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1980. xvi+437 pp. [SDLG] Y. Shi, M. Dehmer, X. Li, I. Gutman, Graph Polynomials, CRS Press, Taylor and Francis Group; A Chapman and Hall Book, Discrete Mathematics and Its Applications series, 2017. [Shu1] A. N. Shumakovitch, Torsion of Khovanov homology. Fund. Math. 225 (2014), no. 1, 343–364. arXiv:math/0405474 [math.GT]. [Shu2] A. N. Shumakovitch. “KhoHo-A Program for Computing and Studying Khovanov Homology.”İ http://github.com/AShuma kovitch/Khoho. [Sik1] A. S. Sikora, On Conway algebras and the Homflypt polynomial, J. Knot Theory Ramifications 6(6), 1997, 879–893. [Sik2] A. S. Sikora, Skein modules and TQFT. Knots in Hellas ’98 (Delphi), 436–439, Ser. Knots Everything, 24, World Sci. Publ., River Edge, NJ, 2000. [Sik3] A. S. Sikora, SLn -character varieties as spaces of graphs. Trans. Amer. Math. Soc. 353 (2001), no. 7, 2773–2804. arXiv:math/9806016 [math.RT]. [Sik4] A. S. Sikora, Skein theory for SU(n)-quantum invariants. Algebr. Geom. Topol. 5 (2005), 865–897. arXiv:math/0407299 [math.QA]. [Sil] D. S. Silver, The last poem of James Clerk Maxwell. Notices Amer. Math. Soc. 55 (2008), no. 10, 1266–1270. [Sim] R. Simion, A type-B associahedron. Formal power series and algebraic combinatorics (Scottsdale, AZ, 2001). Adv. in Appl. Math. 30 (2003), no. 1–2, 2–25. [Simo] O. Simony, Über eine Reihe neuer Thatsachen aus dem Gebiete der Topologie, Math. Annalen, 19 (1882), 110–120, and 24 (1884), 253–280. [Sin] J. Singer, Three-dimensional manifolds and their Heegaard diagrams. Trans. Amer. Math. Soc. 35 (1933), no. 1, 88–111. [Sma1] S. Smale. Diffeomorphisms of the 2-sphere. Proc. Amer. Math. Soc., 10:621–626, 1959. [Sma2] S. Smale, The generalized Poincaré conjecture in higher dimensions. Bull. Amer. Math. Soc. 66 (1960), 373–375. [Sma3] S. Smale, Generalized Poincaré’s conjecture in dimensions greater than four. Ann. of Math. (2) 74 (1961), 391–406.

References

497

[Sma4] S. Smale, On the structure of manifolds. Amer. J. Math. 84 (1962), 387–399. [Smit] The Smith Conjecture, Edited by J. W. Morgan and H. Bass. New York Academic Press 1984. [Sta] P. Stackel, Gauss als Geometer, in the X’th Volume of Gauss’ Collected Works. [Stal] J. R. Stallings, A topological proof of Grushko’s theorem on free products. Math. Z. 90 (1965), 1–8. [Stan1] R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics,and geometry, Ann. New York Sci., 576, New York Acad. Sci., New York,1989, 500–535. [Stan2] R. P. Stanley, Catalan numbers. With history of the Catalan numbers by Igor Pak. Cambridge University Press, New York, 2015. viii+215 pp. [Stan3] R. P. Stanley, Enumerative combinatorics. Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin, Cambridge Stud. Adv. Math., 62 Cambridge University Press, Cambridge, 1999. xii+581 pp. [Ste] E. Steinitz, Beiträge zur Analysis situs. Sitz.-Ber. Berl. Math. Ges. 7, 29–49 (1908). [Sti] E. Stiefel, Richtungsfelder und Fernparallelismus in ndimensionalen Mannigfaltigkeiten. (German) Comment. Math. Helv. 8 (1935), no. 1, 305–353. [Stu] M. A. Stukow, A finite presentation for the mapping class group of a nonorientable surface with Dehn twists and one crosscap slide as generators. J. Pure Appl. Algebra 218 (2014), no. 12, 2226–2239. [SCKSSWW] D. W. Sumners (Editor), N. R. Cozzarelli, L. H. Kauffman, J. Simon, D. W. Sumners, J. H. White, S. G. Whittington, New Scientific Applications of Geometry and Topology, Proceedings of Symposia in Applied Mathematics, Vol. 45, AMS 1992. [Sus] A. K. Sushkevich - The theory of generalized groups, Harkov-Kiev, ONTI, 1937 (in Russian: Teoriya obobshchennyh grupp). [Syl] J.J. Sylvester, Proof of the hitherto undemonstrated fundamental theorem of invariants, The Collect. Math. Papers of James Joseph Sylvester, vol. 3, Chelsea, New York, 1973, pp. 117–126. [Tai1] P. G. Tait, Some elementary properties of closed plane curves, Messenger of Mathematics, New Series, No.69, 1877, (communicated at the 1876 Meeting of the British Association). Reprinted in Scientific Papers of P. G. Tait, 270–272.

498

References

[Tai2] P. G. Tait, Johan Benedict Listing, Nature, February 1, 1883. Reprinted in Scientific Papers of P.G.Tait. [Tai3] P. G. Tait, On knots I, II, III, Scientific Papers, Cambridge University Press, 1898–1900. Including: Trans. R. Soc. Edin., 28, 1877, 35–79. Reprinted by Amphion Press, Washington D.C., 1993. [Tau] C. F. Taubes, Gauge theory on asymptotically periodic 4-manifolds. J. Differential Geom. 25 (1987), no. 3, 363–430. [TL] H. Temperley and E. Lieb, Relations Between the ‘Percolation’ and ‘Colouring’ Problem and Other Graph-Theoretic Problems Associated with Regular Plane Lattices: Some Exact Results for the ‘Percolation’ Problem, Proceeds of the Royal Society of London 322 (1971), 251–280. [Thi1] M. B. Thistlethwaite, Knot tabulations and related topics, Aspects of topology, LMS Lecture Notes Series, 93 (1985), 1–76. [Thi2] M. B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology, 26 (1987), 297–309. [Thi3] M. B. Thistlethwaite, Kauffman’s polynomial and alternating links. Topology 27 (1988), no. 3, 311–318. [Thom] E. Thomas, Cross-sections of stably equivalent vector bundles, Quart. J. Math. Oxford Ser. (2) 17 (1966), 53–57. [Thoms] W. Thomson (1867) On Vortex Atoms, Proceedings of the Royal Society of Edinburgh 6: 94–105. [Thu] D. P. Thurston, Positive basis for surface skein algebras. Proc. Natl. Acad. Sci. USA 111 (2014), no. 27, 9725–9732. arXiv:1310.1959 [math.GT]. [Thur1] W. P. Thurston, The Geometry and Topology of 3-manifolds, Mimeographed notes, 1977–1979. [Thur2] W. P. Thurston, Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. [Tie1] H. Tietze, Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten. Monatsh. für Math. und Phys. 19, 1–118 (1908). John Stillwell translated the paper: On the topological invariants of multidimensional manifolds; [Tie2] H. Tietze, Über stetige abbildungen einer quadrätflache auf sich selbst, Rend. Circ. Mat. Palermo vol. 38 (1914) pp. 247–304. [Toy] K. Toyoda, On affine geometry of abelian groups, Proceedings of the Imperial Academy, Volume 16 (5), 1940, 161–164.

References

499

[Tra1] P. Traczyk, 10101 has no period 7: a criterion for periodic links, Proc. Amer. Math. Soc., 180, 1990, 845–846. [Tra2] P. Traczyk, A criterion for knots of period 3, Topology and its Appl., 36 (3), 1990, 275–281. [Tra3] P. Traczyk, Periodic knots and the skein polynomial, Invent. Math., 106 (1), 1991, 73–84. [Trac] P. Traczyk, A new proof of Markov’s braid theorem. Knot theory (Warsaw, 1995), 409–419, Banach Center Publ., 42, Polish Acad. Sci. Inst. Math., Warsaw, 1998. [Tra5] P. Traczyk, A combinatorial formula for the signature of alternating diagrams, Fund. Math., 184, 2004, 311–316. [Tral] L. Traldi, On the Goeritz matrix of a link, Math. Z., 188, 1985, 203–213. [Tri] A.G. Tristram, Some cobordism invariants for links, Proc. Combridge Phil. Soc., 66, 1969, 251–264. [Tro] H. F. Trotter, Non-invertible knots exist. Topology, 2, 1963, 275– 280. [Tsu] T. Tsukamoto, The almost alternating diagrams of the trivial knot, Journal of Topology, 2,2009, no. 1, 77–104. arXiv:math/0605018 [math.GT]. [Tu] Loring W. Tu, An introduction to manifolds. Second edition. Universitext. Springer, New York, 2011. xviii+411 pp. [Tur1] V. G. Turaev, A simple proof of the Murasugi and Kauffman theorems on alternating links. Enseign. Math. (2) 33 (1987), no. 3–4, 203–225. [Tur2] V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math., 92, 1988, 527–553. [Tur3] V. G. Turaev, The Conway and Kauffman modules of a solid torus. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988), Issled. Topol. 6, 79–89, 190; translation in J. Soviet Math. 52 (1990), no. 1, 2799–2805. [Tur4] V. G. Turaev, Algebras of loops on surfaces, algebras of knots, and quantization. Braid group, knot theory and statistical mechanics, II, 324–360, Adv. Ser. Math. Phys., 17, World Sci. Publ., River Edge, NJ, 1994. [Tur5] V. G. Turaev, Skein quantization of Poisson algebras of loops on surfaces. Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 6, 635–704. [TuSi] R. E. Tuzun, A. S. Sikora, Verification of the Jones unknot conjecture up to 24 crossings. J. Knot Theory Ramifications 30 (2021), no.3, Paper No. 2150020, 6 pp. arXiv:2003.06724 [math.GT].

500

References

[Van] A. T. Vandermonde, Remarques sur les problemes ` de situation, Memoires ` de l’Academie ` Royale des Sciences (Paris) (1771), 566– 574. [Vas] G. Vasari, Lives of the Most Excellent Italian Painters, Sculptors, and Architects, from Cimabue to Our Times, Florence 1550, Modern Edition: The Loves of the Artists (Oxford World’s Classics). Oxford University Press, 1998. ISBN 0-19-283410-X Chapter: LIFE OF LEONARDO DA VINCI: Painter and Sculptor of Florence. [VW] O. Veblen, J. H. C. Whitehead, A set of axioms for differential geometry. Proceedings of the National Academy of Sciences 17 (1931), pp. 551–561. The Mathematical Works of J. H. C. Whitehead, Vol. 1, Pergamon Press, 1962, pp. 93–104. [Vev] M. Veve, Torsion in the KBSM of a 3-manifold caused by an incompressible torus and detected by a complete hyperbolic structure. Kobe J. Math. 16 (1999), no. 2, 109–118. [Vir1] O. Viro, Remarks on definition of Khovanov homology, e-print: arXiv:math/0202199 [math.GT]. [Vir2] O. Viro, Khovanov homology, its definitions and ramifications, Fund. Math., 184, 317–342, 2004. [Vog] H. Vogt, Sur les invariants fondamentaux des équations différentielles linéaires du second ordre. (French) Ann. Sci. École Norm. Sup. (3) 6 (1889), 3–71. [Vre] A. Vretblad, Fourier Analysis and its Applications, Graduate text in Mathematics, 223, Springer, 2003. [Wal1] F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II. (German) Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117. [Wal2] F. Waldhausen, On the determination of some 3–manifolds by their fundamental groups alone, Proc. of International Symposium in Topology, Hercy-Novi, Yugoslavia, 1968, Beograd (1969) 331–332. [Walk] K. Walker, On Witten’s 3-manifold invariants. preprint (1991). [Wall] A. H. Wallace, Modifications and cobounding manifolds, Can. J. Math. 12, 1960, 503–528. [Wan] X. Wang, PhD Thesis: Knot Theory and Algebraic Structures Motivated by and Applied to Knots, 2018, George Washington University. [Wat] T. Watanabe, Some exotic nontrivial elements of the rational homotopy groups of Diff(S 4 ). Preprint December 2018, arXiv:1812.02448 [math.GT]

References

501

[Wei] N. M. Weinberg, On free equivalence of free braids, C.R. (Dokl.) Acad. Sci. USSR, 23 (1939) pp. 215–216 (in Russian). [Wen] H. Wenzl, On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), no. 1, 5–9. [Wes] B. W. Westbury, The representation theory of the Temperley-Lieb algebras. Math. Z. 219 (1995), no. 4, 539–565. [Whi1] J. H. C. Whitehead, A certain open manifold whose group is unity. The Quarterly Journal of Mathematics, Volume os-6, Issue 1, 1935, Pages 268–279. [Whi2] J. H. C. Whitehead, On C 1 -complexes. Ann. of Math. (2) 41 (1940), 809–824. [Whi3] J. H. C. Whitehead, On incidence matrices, nuclei and homotopy types. Ann. of Math. (2) 42 (1941), 1197–1239. [Whi4] J. H. C. Whitehead, Manifolds with transverse fields in euclidean space. Ann. of Math. (2) 73 (1961), 154–212. [Whi5] J. H. C. Whitehead, The immersion of an open 3-manifold in euclidean 3-space, Proc. London Math. Soc. (3) 11 (1961), 81–90. [Whit] H. Whitney, The coloring of graphs, Ann. of Math., 33, 1932, 688– 718. [Win] A. Wintner, Asymptotic distributions and infinite convolutions, Edwards Brothers, Ann Arbor, Michigan, 1938. [Wir] W. Wirtinger, Uber die Verzweigungen bei Funktionen von zwei Veranderlichen, Jahresbericht d. Deutschen Mathematiker Vereinigung, 14 (1905), 517. (The title of the talk supposedly given at September 26 1905 at the annual meeting of the German Mathematical Society in Meran). [Wit] E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351. [WK] D. Wolkstein, S. N. Kramer, Inanna: Queen of heaven and earth: Her Stories and Hymns from Sumer, Harper Collins Publisher, 1983. [Wu1] Y-Q. Wu, Jones polynomial and the crossing number of links, Differential geometry and topology (Tjanjin, 1986–87), Lectures Notes in Math., 1369, Springer, Berlin - New York, 1989, 286–288. [Wu2] Y-Q. Wu, Cyclic surgery and satellite knots, Topology Appl. 36 (1990), 205–208. [Yam1] S. Yamada, A topological invariant of spatial regular graphs, in: Proc. Knots 90, De Gruyter, Berlin, 1992, pp. 447–454. [Yam2] S. Yamada, How to find knots with unit Jones polynomials, in Knot Theory, Proceedings of the conference dedicated to Professor

502

References

Kunio Murasugi for his 70th birthday, Toronto, July 13th-17th, 1999, March 2000, 355–361. [Yok1] Y. Yokota, The Jones polynomial of periodic knots, Proc. Amer. Math. Soc. 113(3), 1991, 889–894. [Yok2] Y. Yokota, The skein polynomial of periodic knots, Math. Ann., 291 (1991), 281–291. [Yok3] Y. Yokota, The Kauffman polynomial of periodic knots, Topology, 32, 1993, 309–324. [Yok4] Y. Yokota, Polynomial invariants of periodic knots, Journal of Knot Theory and its Ramifications, 5(4), 1996, 553–567. [Zho] J. K. Zhong, The Kauffman skein module of a connected sum of 3-manifolds. Topology Appl. 139 (2004), no. 1–3, 113–128. arXiv:math/0205131 [math.GT].

Select Hints and Solutions

Problems of Lecture 1 1.5.6 Answer: .τ(θ n1,n2,...,ns ) =

n

i=1 n1 n2 ...ni−1 ni+1 ...ns .

Problems of Lecture 3 φ

3.1.19 Hint: Show that the map .{a, b, |aba = bab} − → SL(2, Z) given by  .

φ(a) =

 11 , 01

 φ(b) =

1 0 −1 1



is an epimorphism and .φ(a)φ(b)  φ(b)φ(a).

Problems of Lecture 4 4.3.4 Hint: Use Reidemeister moves in a way that changes the role of black and white colors. 4.5.6 Hint: Use the connection between the determinant and the signature, e.g., Det(L) = i σ(L) |Det(L)|; see [Gi, Prz21]. 4.9.6 Hint: Apply Lemma 4.7.3.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5

503

504

Select Hints and Solutions

¯ = −σψ (L). 4.9.8 Hint: Show that σψ ( L)

Problems of Lecture 5 5.2.4 (iii) Answer: .VH+ (t) = −t 2 (t + t −1 ) and .V3¯ 1 (t) = −t 4 + t 3 + t. 3

+1 . (iv) Answer: .V41 (t) = t −2 tt+1 5

5.2.8 Answer: W41 (t) = −t −2 . 5.3.1 Hint: Show that the definition does not depend on the ordering of crossings of D, that is, the order in which crossings are resolved. 5.3.2 Hint: Use the third Tait-flype conjecture (see Lecture 2). 5.6.1 Hint: Resolve all crossings of diagrams by the Kauffman bracket skein relation, and compare results. 5.7.8 Hint: Show that these links have different 3rd Burnside groups (see [DP1] for a solution). 5.8.1 Answer: V31 (t) = −t −4 + t −3 + t −1 . 5.8.2 Hint: Refer to Exercise 5.2.4 (ii) and the chirality of the figure-eight knot. 5.8.4 Hint: Check your calculations with [KAT].  = sw(D) + 2l k( D),  where D  is an oriented diagram obtained by 5.8.7 Hint: w( D) choosing any fixed orientation on an unoriented diagram D.

Problems of Lecture 6 6.2.4 Hint: Check the initial conditions, and then use skein relation in the inductive step. 6.2.7 Hint: Compare to Exercise 5.4.6 part (b) from the previous lecture. 6.5.8 Answer: Λ41 (a, z) = (a + a−1 )(z 2 − 1)(z + a + a−1 ) + 1.

Select Hints and Solutions

505

6.5.9 Hint: Use Theorem 5.4.4 and Lickorish’s observation [Lic4] that FL (1, −1) = (−1)com(L)VL2 (e πi/3 ); compare [Prz24]. 6.5.16 Hint: We have

ia+(ia)−1 −iz

 −1=−

a−a−1 z

 +1 .

6.7.3 Answer: (ii) P72 = −a−8 − a−6 − a−2 − a−4 z 2 + a−6 z 2 + a−2 z 2 .

Problems of Lecture 7 7.2.11 Hint: Use the method of the proof of Corollary 7.2.7. 7.2.12 Hint: Notice that n s-legged spiders cut the disk into 1 + (s − 1)n regions.

14

2n



n−2 2n 7.2.14 Answer: C6,(3) = 2 13 4 , C7,(3) = 9 4 , Cn,(1) = n−2 , Cn,(2) = 2 n−2 . 7.4.2 Hint: There are

6

2

distinct states.

Problems of Lecture 8 8.4.14 Answer: . A3 + A(e1 + e2 ) + A−1 (e1 e2 + e2 e1 ). 8.5.6 Answer: σ1 σ22 .

Problems of Lecture 9 9.5.8 Hint: See Lecture 10 where a generalization of Exercise 9.5.8 is considered using the plucking polynomial; compare [DaLiPr, DP3].

Problems of Lecture 10 10.2.8 Answer: The proof is by induction on n. For the base case .n = 1, since 1

1

1

 1 1 i 1−i . Assume that the formula holds for . 1 q = 1 = 0 q , then .(x + y) = i qx y i=1

506 .

Select Hints and Solutions

n − 1; then (x + y)n = (x + y)(x + y)n−1  n−1   n−1 = (x + y) x i y n−i−1 i, n − i − 1 q i=0    n−1 n−1    n−1 n−1 = x i+1 y n−i−1 + yx i y n−i−1 . i, n − i − 1 i, n − i − 1 q q i=0 i=0

.

We use . yx = qxy and then shift the i index in the first set of summations to obtain    n−1  n−1 n−1 i n−i i .(x + y) = x y + q x i y n−i i − 1, n − i q i, n − i − 1 q i=0 i=0      n−1   n−1 n−1 i = +q x i y n−i . i, n − i − 1 i − 1, n − i q q i=0 n

n−1  

The result follows after applying Equation 10.1 with .a = i and .b = n − i. 10.2.27 Hint: Use the state sum formula stated in Corollary 10.2.20. 10.3.1 Hint: The solution can be found in [DP3, Prz25]. 10.4.5 Hint: The Jones polynomial of the Whitehead link (with negative selfcrossing, 521 ) is −7/2 .t − 2t −5/2 + t −3/2 − 2t −1/2 + t 1/2 − t 3/2 .

Problems of Lecture 11 11.11.2 Hint: Express unoriented link diagrams as linear combinations of oriented ones. 11.11.3 Hint: Use the method from Lecture 6 where it is shown that the Kauffman and Dubrovnik polynomials are equivalent.

Problems of Lecture 12 12.7.3 Hint: Use the formula for n-moves in the Kauffman bracket polynomial in Lecture 5.

Select Hints and Solutions

507

12.7.8 Hint: L1 = Az + A−1 xy. 12.7.11 Hint: Notice that φ(L (1) + A3 L) = 0 and that φ(L+ − AL0 − AL∞ ) = φ(L+ + L0 + L∞ ). Furthermore, exactly one of the three links L+, L0 , and L∞ has more (by one) components than the other two and, thus, twice as many possible orientations as well. So they cancel in H1 (M, Z) and φ(L+ + L0 + L∞ ) = 0.

Problems of Lecture 13 13.6.1 Hint: Observe that the relations stated above reduce every link in .F1,0 and F1,1 to a linear combination of the basic elements.

.

13.6.3 Hint: Use the Dirac trick described in Lecture 12. 13.6.4 Hint to (2): Use the fact that the center of S alg (Fg,d ) is generated by all the boundary parallel curves. See Theorem 13.4.3. 13.6.5 Hint: Observe that when A = −1, S alg (F0,4 ) and S alg (F1,2 ) are isomorphic. See Exercise 13.6.4 and Remark 13.3.9. 13.6.6 Hint to (1): Use Theorem 13.5.1. Hint to (2): The relation from Fig. 13.11 leads to the Chebyshev-type relation cn x = A−1 cn+1 + Acn−1 . From this, conclude that cn = An−1 c1 Sn−1 (x) − An c0 Sn−2 (x).

Problems of Lecture 15 15.7.1 Hint: Use the Seifert surface. 15.7.2 Hint: Let T 2 = ∂ M . Prove that there is a curve on T 2 that is homologically trivial in M . This curve is sometimes called the standard rational framing of the knot.

Problems of Lecture 16 16.2.7 Hint: Use .Δn = (−1) sum is a geometric series.

n (A2n+2 −A−2n−2 )

A2 −A−2

and . A4r = 1, and then recognize that the

508

Select Hints and Solutions

Problems of Lecture 17 Problems of Lecture 18 18.3.2 Hint: Notice that .

2n

 2n k=0

k

= 22n and .

2n

k

=



2n

2n−k .

18.4.4 Hint: Consider a class of annular connections resulting from placing the singularity in each region of a Catalan connection in D2 . Let A1 and A2 be two elements of the same class. For any b ∈ Mbn , what can be said about the exponents of x, y, and w for B, A1  and B, A2 ?

Problems of Lecture 19 ¯ = H −a,−b (D), and use the universal coefficient 19.6.4 Hint: Notice that .Ha,b ( D) theorem for cohomology.

Problems of Lecture 20 20.1.8 Hint: Notice that if the state s is different from the state .s A by one label at a crossing, then .σ(s) = |cr(D)| −2 and .σ(s)+2τ(S) < |cr(D)| +2|Ds A | −4. Therefore, there is only one enhanced Kauffman state with grading .b = |cr(D)| + 2|Ds A |. 20.1.10 Hint 1: Notice that if D is an alternating diagram, then after smoothing any crossing, the diagram remains alternating. The crossing can be chosen so that the resulting diagrams D A and DB contain no nugatory crossings. Hint 2: Show that for connected alternating diagram D, we have .

|Ds A | + |DsB | = |E(G)| − |V(G)| + 2 = cr(D) + 2,

where G is a 4-valent graph, a projection of D in S 2 . 20.1.11 Hint: Consider the path v1, v2, ..., vk , ..., and observe that in homology v1 = −v2 = v3 = −v4 · · · = (−1)k−1 vk .

Select Hints and Solutions

From left to right: Deborah, Dionne, Gabriel, Rhea Palak, and Józef

509

Index of Names

A Adams, C. C., 79 Alexander, J. W., 19, 28, 29, 48, 60, 116, 163, 373, 382, 392, 393 Alexandroff, P. S., 374 Archimedes, 3 Artin, E., 163, 360 Asaeda, M., 359

B Bailey, J., 425 Bakshi, R. P., 241, 265, 283, 284, 286, 311, 323 Bankwitz, C., 19 Bar-Natan, D., 225, 359, 361 Barad, G., 323 Barrett, J. W., 235, 284, 289 Baxter, R. J., 49, 155, 169 Birkhoff, G. D., 29 Birman, J. S., 34, 93, 94 Blanchet, C., 296 Boeddicker, O., 9 Bonahon, F., 20 Bousseau, P., 282 Brahana, H., 386 Brandt, R. D., 130 Briggs, G. B., 19, 28 Brody, E. J., 383 Brunn, H. K., 9 Bullock, D., 236, 239, 240, 253, 257 Burstin, C., 129 Burton, B. A., 20

C Cai, X., 169, 311, 312, 314 Cairns, S. S., 376 Carrega, A., 238 Carter, J. S., 55 Caskey, J. L., 1 Chebyshev, P. L., 137, 140 Chen, Q., 323, 324, 329 Chernov, V., 210, 284 Conway, J. H., 19, 60, 72, 88, 110, 115, 158 Crowell, R. H., 38 Culler, M., 258 D da Vinci, L., 1, 4 Dabkowski, M. K., 112, 185, 196, 238, 323 Dasbach, O., 107 Dehn, M., 19, 24, 26, 27, 386 Descartes, R., 16 Detcherry, R., 238 Di Francesco, P., 311, 312 Donaldson, S. K., 376 Dowker, H., 20 Drinfeld, V., 50 Dürer, A., 1, 4 E Ehler, C. L. G., 5 Eilenberg, S., 55 Epstein, D. B. A., 393 Etherington, M. H., 129 Euler, L., 1, 5 Evans, D. E., 155

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5

511

512 F Fenn, R., 55, 410 Fock, V., 261, 266 Foster, R. M., 33 Fox, R. H., 18, 33, 38, 60, 75 Francis I., 3 Frankl, F., 59, 74 Freedman, M. H., 376 Frenkel, I., 335 Freyd, P., 117 Fricke, R., 254 Frohman, C., 254, 265, 266 Fuss, N., 144 G Gabrovšek, B., 216 Gauss, C. F., 1, 7, 9, 38, 80, 365 Gelca, R., 254, 265, 266 Giller, C. A., 72, 88 Gilmer, P. M., 217, 238 Goeritz, L., 59, 60 Goncharov, A., 261, 266 Gordon, C., 63, 67 Green, J., 359 Gunningham, S., 240 Gutman, I., 140 H Habegger, N., 296 Haken, W., 386, 387 Harary, F., 42 Harris, J. M., 240 Hartley R., 127 Haseman, M. G., 19, 26 Hatcher, A. E., 392 Heegaard, P., 19, 23, 26, 27, 379, 386 Heliodorus, 2 Helmholtz, H., 15 Heraklas, 2 Hirsch, M. W., 376 Ho, C. F., 130 Hochschild, G. P., 55 Hopf, H., 54, 374 Hoste, J., 20, 117, 199, 210, 215, 237, 284 Humphries, S. P., 398 Hurewicz, W., 55 Huygens, C., 5

Index of Names Jeffrey, L. C., 304 Johannson, K., 389 Jones, V. F. R., 34, 60, 93, 96, 116, 155, 169, 171 Jordan, D., 240 Joyce, D., 48 Julian the Apostate, 3 K Kaiser, U., 209, 223, 227 Kamada, S., 55 Kassel, C., 156 Kauffman, L. H., 22, 35, 38, 42, 60, 73–75, 88, 93, 95, 98, 118, 132, 134, 155, 158, 336 Kawauchi, A., 113, 127 Kervaire, M. A., 375, 392 Khovanov, M., 335, 341 Kidwell, M., 215 Kirby, R. C., 303, 407, 409 Kirchhoff, G. R., 9, 10 Kirkman, T. P., 18, 26 Klein, F., 254 Kneser, H., 386 Knott, C. G., 16 Ko, K. H., 311 Kovalevskaya, S., 19 Kowitz, K., 152 Kramer, S. N., 1 Kühn, H., 5

I Ibarra, D., 26, 283, 284, 286, 311, 323, 328, 422

L Lambropoulou, S., 216 Landry, A., 201 Lascaris, J., 3 Lê, T. T. Q., 239 Lee, T. D., 49 Leibniz, G. W., 1, 3, 5 Leonard, L., 140 Levine, J., 87 Lewark, L., 359 Li, C., 185 Lickorish, W. B. R., 79, 117, 130, 134, 137, 212, 295, 296, 300, 303, 305, 311, 312, 323, 394, 399, 505 Lieb, E., 155 Listing, J. B., 9 Litherland, R. A., 67 Little, C. N., 18 Lo Faro, W., 239 Lombardero, D. A., 20 Lord Kelvin, 16

J Jaco, W. H., 389

M Magnus, W., 24, 34

Index of Names Manolescu, C., 377 Marché, J., 236, 240 Markov, A., 163, 164 Martin, P., 323 Masbaum, G., 296 Mattman, T. W., 42 Matumoto, T., 407 Maxwell, J. C., 8, 15, 28 Mayer, W., 129 McCool, J., 398 McCullough, D., 210, 284, 286 McMillan Jr., D. R., 238 Melvin, P., 303 Menasco, W., 22 Michelson, A., 17 Millett, K. C., 117, 130, 212 Milnor, J. W., 375, 392 Möbius, A. F., 377, 386 Moise, E. E., 374 Montesinos, J., 38, 112 Montoya-Vega, G., 283, 284, 286, 328 Morley, E., 17 Morrison, S., 359 Morton, H., 110, 118, 216 Mroczkowski, M., 216, 238 Mukherjee, S., 265, 311, 323, 359 Muller, G., 261 Murasugi, K., 22, 72, 87, 122, 126 N Nakanishi, Y., 112, 113 Newton, I., 3, 188, 189, 191 Nicetas, 3 Nielsen, J., 393 Nowiński, K., 130 O Ocneanu,A., 117 Oribasius of Pergamum, 3 Ozsvath, P., 73 P Pacioli, L., 1, 4 Papakyriakopoulos, C. D., 374 Peirce, C. S., 55 Perelman, G., 375 Perko, K. A., 18–20, 425 Poincaré, J. H., 18, 23, 375 Pontrjagin, L., 59, 74 Primaticcio, F., 3 Przytycki, J. H., 72, 93, 112, 117, 122, 129, 130, 185, 205, 237, 265, 283, 284, 286, 311, 323, 324, 360 Purcell, J., 402

513 Q Queffelec, H., 266 R Radó, T., 374 Reidemeister, K., 19, 27–29, 382 Reshetikhin, N. Y., 296, 311 Riemann, B., 365 Roberts, J. D., 295, 305, 410 Roberts, J. H., 392 Rolfsen, D., 18, 133, 425 Rong, Y., 284 Roth, A., 425 Rourke, C., 55, 410 Russell, H., 227 Rutherford, E., 17 S Safronov, P., 240 Saito, M., 55 Saleur, H., 323 Samuelson, P., 216 Sanderson, B., 55 Sazdanović, R., 360 Schütz, D., 359 Scott, C. A., 19 Seifert, H., 59, 60, 74, 388 Shalen, P. B., 258, 389 Shumakovitch, A., 359 Siebenmann, L., 19, 20 Sikora, A. S., 130, 256, 295, 305 Silvero, M., 265, 360 Simion, R., 323, 326 Simony, O., 23 Smale, S., 377, 392 Smolinsky, L., 311 Sokolov, M. V., 9 Solis, P., 42 Steinitz, E., 374 Sushkevich 129 Szabo, Z., 73 T Tait, P. G., 16, 26, 60, 61, 63, 70, 108 Tchebycheff, P. L. (also see Chebyshev), 137 Teller, E., 49 Temperley, H., 155 Thistlethwaite, M., 20, 22 Thomson, J. J., 17 Thomson, W. (see Lord Kelvin), 16 Thurston, D., 266 Tietze, H. F., 23, 374 Traczyk, P., 72, 93, 117, 122, 126, 127, 129, 130

514 Traldi, L., 62, 68 Trotter, H. F., 19 Tsukamoto, T., 92 Turaev, V. G., 107, 156, 205, 215, 265, 284, 296, 311 V Vandermonde, A. T., 6 Vasari, G., 4 Veblen, O., 29, 60, 365 Vidus Vidius, 3 Vogel, P., 296 W Waldhausen, F., 284, 380, 389 Walker, K., 296 Wang, X., 265, 360 Watanbe, T., 392 Weeks. D. E., 283, 284, 286 Weeks, J. R., 20

Index of Names Weinberg, N. M., 163 Wenzl, H., 169, 298 Westbury, B., 311, 312 Whitehead, J. H. C., 365, 375 Wirtinger, W., 24 Witten, E., 236, 295, 296, 301, 311 Wolf, C., 6 Wolff, M., 238 Wolkstein, D., 1 Y Yamada, S., 199 Yang, C. N., 49 Yang, S. Y., 360 Yasuhara, A., 383 Yetter, D., 117 Z Zhong, J. K., 217 Zilber, J. A., 55

Subject Index

Symbols 2-handle, 218, 232, 244 3-handle, 218, 232, 244 A-symmetrizer, 171, 174 R P 3 # R P 3 , 241 SL(2, C) character variety, 257–259 Δ-equivalence, 28 Δ-move, 27 θ-curve, 11 2-bridge knot, 23, 239, 240 3-manifold, 35, 136, 196, 205, 206, 298–301, 305, 306, 391, 399, 400, 403, 410–412 E8 manifold, 376 lens space, 256, 381 ˆ I , 216, 237, 256 R P2 × R P 3 , 136, 256 F0,3 × S 1 , 238 T 3 , 238 P2 -irreducible, 388 S 1 × S 2 , 217, 237, 287, 380, 385, 386, 388, 402 Haken, 387, 388 integral homology sphere, 283, 400, 403 invariant, 295, 296, 299, 301, 311, 322, 323 irreducible, 243, 385 lens space, 216, 237–239, 304, 380–383, 402, 403, 405, 409 Poincaré homology sphere, 213, 389, 402, 403 prime, 234, 243, 384 prism, 238 quaternionic, 238, 389

rational homology sphere, 213, 283, 294 Seifert fibered, 227, 380, 388, 389 Whitehead manifold, 237, 375 A abstract algebra, 127 achirality, 133 action of a group, 122 acyclic, 186 adequate, 97, 103–106 adequate link, 352 Alexander-Conway polynomial, 59, 92, 115 Alexander matrix, 32 Alexander module, 31 Alexander numbering, 33 Alexander polynomial, 23, 31, 34, 60, 93, 115, 116, 126 alternating diagram, 18, 22 knot, 18 link, 352, 355, 360 alternating diagram almost alternating diagram, 92 ambient isotopy, 37, 131, 132, 158, 207, 307, 312, 401, 409 amphicheiral, 9, 19, 20, 24, 92, 119, 126, 198 anti-symmetrizer, 170, 171 arrow diagrams, 241 Artin braid group, 155, 163, 165, 169, 171, 360, 362 atlas, 366 atoroidal, 243, 387

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 J. H. Przytycki et al., Lectures in Knot Theory, Universitext, https://doi.org/10.1007/978-3-031-40044-5

515

516 B bigrading, 338 billiard knot, 127 Bin(X), 128 binary operation, 46 binary tree, 195 blackboard framing, 296, 400, 407 Boltzmann weight, 51, 194 braid, 1, 6–8, 34, 41, 46, 48, 112, 155, 163, 164, 166, 171, 173, 176, 359–362 braid axis, 164, 165 braid index, 359, 361 Brandt-Lickorish-Millett-Ho polynomial, 132, 134 Brauer algebra, 224 bridge number, 42 Burnside group, 113 Burnside group of a link, 112 C Catalan connections, 319, 324, 326 Catalan number, 157, 162, 163, 260 Catalan states, 197, 199–202 category, 231, 371 DIFF, 372, 374, 376, 377 PL, 372, 374, 376, 377 TOP, 371, 374, 376, 377 chain complex, 54, 56, 336, 337, 343, 350 change of root, 194 characteristica geometrica, 5 chart, 366 Chebyshev basis, 255, 266, 267 Chebyshev polynomial, 115, 120, 137–140, 142, 175, 179, 241, 251, 254, 257, 265, 267, 282, 313, 314, 316, 319, 323, 325, 326 chromatic polynomial, 29, 30 cluster algebra, 261 cocore, 378 codimension, 368 colored Jones polynomial, 295–298 coloring, 29, 49, 51 combinatorial topology, 5 composition of knots, 78 compressing disc, 387 computational tree, 126 connected sum of knots, 78 of links, 96, 131, 407 of manifolds, 217, 224, 234, 299, 301, 327, 383, 408 Conway algebra, 115, 127–129 Conway polynomial, 115, 116, 132 Conway relations, 128

Subject Index Conway skein triple, 94, 127 Conway type invariant, 127, 130 core, 378 core group, 113 crossingless connection, 158, 160, 161, 197, 312, 318, 324, 327–329, 331 cycle, 81 cyclic group, 122 cyclotomic polynomials, 194, 238 D Danzig, 5 Dehn filling, 240, 402 Dehn homeomorphism, 283, 285–288, 291 Dehn subgroup, 285 Dehn surgery, 240 Dehn twist, 285, 286, 288, 291, 391, 394, 395, 397–399, 406, 407 delay function, 185, 199–201 descending diagram, 94, 122 determinant of a knot, 59, 62, 65, 67, 80 determinant of a link, 59, 68, 69 Dickson polynomial, 115, 140 diffeomorphism, 371 orientation preserving, 371 orientation reversing, 371 differential map, 338 dihedral branch covering, 112 Dirac trick, 283, 284, 288, 289 disjoint sum, 96 double branched cover, 101, 134 Drinfeld-Turaev quantization, 214 dual graph, 109, 197, 198 dual handle, 379 Dubrovnik polynomial, 115, 134, 136 Dyck paths, 314, 317, 321, 322 E eigenvalue, 299, 300, 411, 412 elliptic Hall algebra, 216 enhanced Kauffman states, 338, 342, 343 entropic, 115, 129 entropy, 127, 128 essential surface, 387 Euler characteristic, 336, 345, 346 exotic sphere, 375 F Farey diagram, 270 Fibonacci numbers, 142 figure-eight knot, 13, 41, 58, 86, 95, 112, 119, 134 five lemma, 234 Four color conjecture, 197

Subject Index Fox coloring, 46, 93, 100, 134 Fox n-coloring, 40, 47, 111 Fox 3-coloring, 38, 39 framed knots and links, 283, 284, 286–288, 290–293 framed link, 98, 118, 207, 296, 298–300, 305, 306, 400, 401, 407–412 framing, 207, 291, 292, 307, 400, 401, 403, 409 number, 400–403, 407, 409–411 surgery, 306, 307, 400, 403 framing relation, 207 Frobenius algebras, 226, 227, 340 fundamental group, 33, 58 G Gaussian polynomial, 188, 189 generalized Jones polynomial, 117 generalized type A inner product, 318 geometria situs, 5 geometric intersection number, 271 geometry of position, 5, 6 Goeritz matrix, 59–63, 67–69, 71, 80, 91 Goldman-Wolpert Lie algebra, 220 Gram determinant, 145, 152, 243, 261, 311 Gram determinant of generalized type A, 318–322 Gram determinant of type A, 169, 312, 314, 319 Gram determinant of type B, 324, 326 Gram determinant of type Mb, 323, 324, 330 Gram matrix, 325, 326, 328, 330, 331, 333 Gram-Schmidt procedure, 139, 312 graph, 186 graph theory, 6, 9 greedy algorithm, 158 Gromov-Witten theory, 282 H handle, 377 handle addition, 377 handlebody, 233, 234, 379 handle decomposition, 378 handle sliding, 137, 232–234, 237, 239, 244, 284, 311 Hauptvermutung, 374 Heegaard genus, 380 Heegaard splitting, 233, 379–381 Hermite polynomial, 140 Hermitian form, 88 Hermitian product, 139 Hochschild homology, 352 homeomorphism, 306, 366, 391, 392, 394, 397–399, 401–403, 405

517 orientation preserving, 369, 391, 393, 394, 397, 407 orientation reversing, 369, 379–383, 392, 401–403, 406 PL, 373, 391, 394 HOMFLYPT polynomial, 35, 56, 93, 111, 115–118, 121, 123, 125, 129, 131, 132, 140 homology group, 23, 52, 54, 344, 353 homotopy group, 23 homotopy sphere, 375 Hopf link, 8, 83, 95, 118, 326 hyperbolic knot, 240 I idempotent, 137, 169–171, 174, 177, 179, 180, 184, 199, 295–299, 303, 305, 312, 314, 316, 323, 326 incidence number, 338 incompressible surface, 227, 387 invariant, 22, 46, 265, 336, 345 Italy, 3 J Jones conjecture, 97 Jones polynomial, 15, 22, 34–36, 39, 62, 93–97, 99–102, 111, 114, 116, 122, 126, 134, 137, 198, 295, 297, 311, 335, 336, 341 Jones skein relation, 99 Jones-Wenzl idempotent, 137, 155, 165, 169, 171, 174, 177, 180, 184, 199, 295–298, 303, 305, 312, 314, 316, 323, 326 JSJ decomposition, 389 K Kauffman bracket polynomial, 73, 93, 97, 98, 101, 111, 121, 131, 134, 155, 166, 196, 201, 295, 296, 300, 311, 336, 345 skein algebra, 138, 251–259, 265, 266, 268, 269 skein module, 171, 196, 197, 210, 223, 230–238, 240–246, 292, 296, 305, 312, 322, 323 skein relation, 171–177, 185, 196, 197, 349 Kauffman-Harary Conjecture, 42 Kauffman polynomial, 100, 111, 115, 130–136, 212 Kauffman states, 337, 342, 357 Kawauchi 4-move conjecture, 113 Khovanov homology, 97, 207, 225, 335–337, 341, 349, 359, 361 Kinoshita-Terasaka knot, 114

518

Subject Index

Kirby calculus, 407, 410 move, 296, 305, 407, 409–412 0 move, 410 1 move, 296, 300, 407, 408, 411, 412 2 move, 298, 300, 305, 407, 409, 411, 412 Klein bottle, 257, 327, 368–370 knot complement, 33 knot group, 33 knot invariant, 32 knot 942 , 119 knots, 1, 6 knot theory, 1–3, 6, 8, 10, 51 Königsberg, 5, 6

Metropolitan Museum of Art, 1 mirror image, 18, 24, 93, 95, 119, 131, 133, 134 mirror symmetry, 282 mixed crossing, 135 Möbius band, 147, 148, 154, 244, 260, 322–324, 327–329, 331, 369, 370, 393, 394, 398, 399, 414 Montesinos-Nakanishi 3-move conjecture, 93, 101, 111, 225 move t¯2k , 121 move tn , 59, 70, 120 multinomial coefficient, 190 Murdoch-Toyoda theorem, 130 mutation, 110, 131, 133

L Laguerre polynomial, 115, 140 lattice crossing, 183–185, 196, 197, 201 lattice path, 157, 317 leaves, 186, 195, 199, 200 Legendre polynomial, 139 Lerna, 1 light bulb trick, see also Dirac trick, 242, 283, 284, 287 link diagram, 28, 31, 47, 51, 134, 158, 338, 341, 349, 400 link homotopy, 219 linking matrix, 299, 300, 391, 411, 412 linking number, 85, 133, 400, 407 link invariant, 358 long exact sequence, 349, 352 Lucas numbers, 142

N Nakanishi 4-move conjecture, 93, 113 n-ary operation, 127 Newton’s binomial formula, 189 Newton’s multinomial formula, 190 n-move, 70, 101 nonalternating knot, 361 nonamphicheiral, 20, 24, 119 non-associative algebra, 129 noncommutative plane, 138 noncommutative torus, 138, 254 non-separating sphere, 209, 210, 213, 217, 242, 283, 285, 287, 288, 385 non-separating torus, 209 normal form, 155, 157, 158, 162, 163 nugatory crossing, 22

M magma, 46, 129 manifold, 365 with boundary, 369 closed, 369 combinatorial, 373 compact, 369 open, 369 orientable, 369, 371, 373 smooth, 370, 371 topological, 366, 368–370 triangulated, 373 mapping class group, 270, 283–286, 391–393, 397–399 Marché’s conjecture, 240 Markov trace, 179 matching polynomial, 140 medial, 129 meridional, 306, 307 Mesopotamia, 1 meta-bracket, 296, 300

O Oberwolfach, 134 orientation preserving embedding, 231 orthogonal polynomial, 139 P palindromic polynomials, 194, 195, 201 Papakyriakopoulos’ Loop theorem, 284 parallelizable, 289, 391, 420–423 periodicity criteria, 123, 127 periodic link, 122, 123, 126 Perko pair, 19 Perko’s notation, 125 permutation group, 155, 165, 171 plane rooted tree, 186, 192, 199, 200 plucking polynomial, 185, 186, 188, 190, 191, 193, 194, 196, 199–201 Poincaré conjecture, 375 Poincaré polynomial, 346 Poisson algebra, 214 polygonal knots, 27 polygonal links, 28

Subject Index polynomial link invariant, 205 positivity conjecture, 281, 282 presimplicial complex, 55 pretzel knot, 86 pretzel link, 101 pre-Yang-Baxter operator, 50 prime decomposition, 385 prime knot, 79 Princeton, 30 product manifold, 367 product-to-sum formula, 139, 254, 255, 272 projection plane, 31 pure braid group, 166 Q q-commutative polynomial, 189 quandle, 51 quantum group, 296 quantum integer, 172, 188, 275 quantum plane, 189 quasi-alternating, 73, 74 quasigroup, 130 quaternion group, 238 R rational knots, 158 real projective plane, 327, 370 Rees algebra, 121 regular isotopy, 98, 118, 132 regular projection, 28 Reidemeister move, 39, 40, 42, 47, 52, 63, 98, 133, 336, 341 relative Kauffman bracket skein algebra, 261 relative Kauffman bracket skein module, 166, 243, 260 relative link diagram, 261 Renaissance, 3 S Seifert matrix, 59, 60, 74, 80–82, 85, 87, 89–91 Seifert surface, 59, 74, 75, 77, 80, 81 self-crossing, 114, 135, 221 self-writhe, 114 semi-simplicial complex, 55 separating sphere, 235 short exact sequence, 350 signature of a link, 59, 67–69 of the linking matrix, 300, 412 of the manifold, 301, 412 Tristram-Levine, 59, 74, 80, 88–91 simplex, 372 simplicial complex, 372, 391 skein, 231

519 skein algebra, 208 Kauffman bracket skein algebra, 138, 251–259, 265, 266, 268, 269 HOMFLYPT skein algebra, 215 homotopy skein algebra, 220 q-homology skein algebra, 213 second skein algebra, 212 signed skein algebra, 208 skein module, 35, 115, 131, 136 Bar-Natan skein module, 207, 226 Dubrovnik skein module, 225 framing skein module, 210, 283, 291 HOMFLYPT skein module, 214 homotopy skein module, 219 Kauffman bracket skein module, 206, 210, 223, 292 Kauffman skein module, 223 kth skein module, 218 q-deformation of the fundamental group, 208 q-homology skein module, 212, 293 q-homotopy skein module, 221 second skein module, 211 signed skein module, 207 Witten skein module, 295, 305, 306, 308 skein polynomial, 117 skein relation, 53, 115, 116, 134, 140, 336 skein triple, 117, 230 Smith Conjecture, 123 smoothing, 337 spin structure, 235, 259, 283, 284, 288–293 square knot, 58 stabilization, 381 state product formula, 192–194 statistical mechanics, 36, 49, 50 strongly adequate, 360 submanifold, 368 subtrees, 192 surgery, 378 Dehn, 378, 383, 403 integral, 239, 299, 378, 400–403, 406, 407 on links, 391 rational, 403 symmetric polynomial, 195 symmetric tensor algebra, 208, 215, 220 symmetrizer, 169–171, 174 T Tait conjectures, 19, 22, 102, 103 Tait-flype, 22, 504 Tait graph, 25 tangent bundle, 284, 289, 290, 292, 413, 415–421 tangent space, 416

520 tangle, 155, 158–160, 163, 166, 171, 177, 178, 199, 203, 206, 207, 224, 225 Temperley-Lieb algebra, 155, 156, 158, 165, 169, 171, 172, 262, 296, 312, 315, 323, 335 theta curve, 314 net, 181, 314 time complexity, 115 Tiryns, 1 Topological Quantum Field Theory, 295, 322 topology, 5, 6, 9, 10, 15, 17, 23, 24, 26, 27, 54, 60, 205, 206, 209, 229, 284, 285, 345, 365, 372, 374, 414, 415 low-dimensional topology, 249, 271 torsion in Khovanov homology, 355, 358 in skein modules, 209, 213, 217, 241, 242 torus knot, 163, 164, 239, 240, 352, 361 torus link, 101, 120, 122 transition map, 366 trefoil knot, 32, 39, 86, 95, 112, 118, 119, 167, 239, 342, 343, 346, 361 triangulation, 373 trigonometric identity, 137 Tristram-Levine signature, 92 trivial knot, 94, 132, 352, 353 trivial link, 111, 133, 136 Troy, 1 Turaev genus, 107 Turaev surface, 107

Subject Index twist knot, 101, 121, 239 U unimodal polynomial, 195, 201 universal algebra, 127 universal enveloping algebra, 221 universal group, 41 universe, 127 unknot, 23, 35, 336 V Vassiliev-Goussarov invariant, 122, 210 Vassiliev invariant, 35 von Neumann algebra, 34, 93, 155 vortex theory, 16 W Warsaw University, 130 wedge product decomposition, 192–194 wedge product of rooted trees, 191, 192 Wirtinger presentation, 24 Witten-Reshetikhin-Turaev invariant, 169, 295, 296, 304, 311, 322, 323, 412 Witten’s conjecture, 213, 236 wrapping number, 246 writhe, 18, 22, 72, 99, 342 Y Yang-Baxter equation, 49, 50, 52 Yang-Baxter homology, 52 Yang-Baxter operator, 52, 56