The mathematics of knots theory and application 9783642156366, 9783642156373, 3642156363, 3642156371

Table of contents :
Contributions in Mathematical and Computational Sciences......Page 5
Preface......Page 7
Contents......Page 9
Introduction......Page 11
Organization......Page 14
Blanchfield and Poincaré Local Systems......Page 15
Passage from Blanchfield Systems to Poincaré Systems......Page 18
Passage from Poincaré Systems to Complex Hermitian Systems......Page 19
Strongly Transverse Poincaré Local Systems......Page 23
Computing Twisted L-Classes for Strongly Transverse Coefficients......Page 25
The Cappell-Shaneson L-Class Formula for Singular Embeddings......Page 26
Nontransverse Coefficient Systems......Page 30
References......Page 39
The Arrow Polynomial......Page 41
Knot 4.01......Page 45
Knot 4.09......Page 49
Knot 4.47......Page 50
Knot 4.99......Page 51
Knots with Arrow Polynomial One......Page 52
References......Page 53
Introduction......Page 54
Conventions and Notation......Page 55
Twisted Reidemeister Torsion......Page 56
Computation of Twisted Reidemeister Torsion......Page 58
Torsion Invariants......Page 59
Twisted Alexander Invariants......Page 60
Relationship Between Twisted Invariants......Page 61
Change of Variables......Page 63
Duality for Twisted Invariants......Page 64
Shapiro's Lemma for Twisted Invariants......Page 65
Twisted Invariants of Knots and Links......Page 66
Distinguishing Knots and Links......Page 67
Twisted Alexander Polynomials of Zero-Surgeries......Page 69
Twisted Alexander Polynomials and Knot Concordance......Page 70
Ribbon Knots and Doubly Slice Knots......Page 73
Twisted Invariants and Slice Links......Page 74
Twisted Alexander Polynomials and Fibered Manifolds......Page 76
Twisted Alexander Polynomials and the Thurston Norm......Page 79
Normalized Twisted Reidemeister Torsion and the Free Genus of a Knot......Page 81
Parabolic Representations......Page 82
Twisted Alexander Polynomials and the Space of 2-Dimensional Representations......Page 83
Metabelian Representations......Page 84
Periodic and Freely Periodic Knots......Page 85
Seifert Fibered Surgeries......Page 87
Homology of Cyclic Covers......Page 88
Definitions and Basic Properties......Page 89
Plane Algebraic Curves......Page 90
Non-commutative Alexander Polynomials......Page 91
Higher Order Alexander Polynomials......Page 92
Comparing Different phi-Compatible Maps......Page 94
Miscellaneous Applications of Higher Order Alexander Polynomials......Page 95
Open Questions and Problems......Page 96
References......Page 97
Introduction......Page 104
The Arrow Polynomial A......Page 106
Khovanov Homology for Virtual Knots......Page 110
Grading Considerations for the Arrow Polynomial A......Page 114
Dotted Gradings and the Dotted Categorification......Page 115
General Setup......Page 118
Part 1. Proof that the Complex is Well Defined......Page 120
Notation......Page 121
Part 2. Proof that the Homology is Invariant Under Reidemeister Moves......Page 128
Explanation for the Second and the Third Moves......Page 129
Applications......Page 130
Open Questions......Page 131
References......Page 132
Introduction......Page 134
The Yokonuma-Hecke Algebra......Page 136
The Adelic Yokonuma-Hecke Algebra......Page 137
Representing the Braid Group......Page 139
The Modular Markov Trace trd......Page 140
The Adelic Markov Trace trinfty......Page 141
Why E-Condition......Page 142
The E-System......Page 143
Lifting Solutions of the E-System......Page 144
Isotopy Invariants from trd......Page 145
Computations......Page 147
A Cubic Skein Relation for Deltad, S......Page 148
An Isotopy Invariant from trinfty......Page 149
References......Page 150
Introduction......Page 152
Main Results......Page 153
Contact Lens Spaces......Page 155
Grid Diagrams......Page 156
The Lagrangian DGA for KL(p,q)......Page 158
The Defect......Page 159
The Boundary Map......Page 161
Converting Fronts to Lagrangian Projections......Page 162
The Lagrangian Projection Associated to a Special Front......Page 163
The Grading......Page 167
Special Boundary Discs......Page 168
Boundary Maps of a-Type Generators......Page 169
Augmentations for (A(K(p,p-1,1)), )......Page 172
Augmentations of (A(K(p,p-1,2)),)......Page 173
References......Page 177
Introduction......Page 178
Atoms and Knots......Page 181
Chord Diagrams, 1-dimensional Surgery and the Kauffman Bracket......Page 185
The Source-sink Condition......Page 191
The Planar Case: Vassiliev's Conjecture......Page 192
The Case of the Klein Bottle......Page 193
The Chord Diagram Algebra and the Graph Algebra......Page 195
Weight Systems Associated with Lie Algebras: a Brief Review......Page 198
Checkerboard Colourable Embeddings......Page 199
Embeddings with Orienting Z2-homology Class......Page 202
The General Case......Page 203
References......Page 204
Introduction......Page 207
Gauss' Formula for Linking Number of Knots......Page 210
Supersymmetry and Morse Theory......Page 211
Graded Algebraic Structures......Page 213
SUSY Quantum Theory......Page 214
Chern-Simons Theory......Page 218
Flat Connections......Page 221
Casson Invariant and Flat Connections......Page 223
Fukaya-Floer Homology......Page 225
Knot Polynomials......Page 228
Categorification of Knot Polynomials......Page 231
Categorification of V(31)......Page 233
Topological Quantum Field Theory......Page 234
Atiyah-Segal Axioms for TQFT......Page 235
Quantum Observables......Page 237
Link Invariants......Page 238
WRT Invariants......Page 241
CFT Approach to WRT Invariants......Page 242
WRT Invariants via Quantum Groups......Page 244
Chern-Simons and String Theory......Page 247
WRT Invariants and String Amplitudes......Page 249
Yang-Mills, Gravity and Strings......Page 252
Gravitational Field Equations......Page 253
Geometrization Conjecture and Gravity......Page 258
References......Page 261
Short Historical Introduction......Page 265
Lattice Knots of Dehn and Heegaard......Page 268
Kirchhoff's Complexity of a Graph......Page 270
Tait's Relation Between Knots and Graphs......Page 272
Link Diagrams and Reidemeister Moves......Page 273
Goeritz Matrix and Signature of a Link......Page 275
Seifert Surfaces......Page 286
Connected Sum of Links......Page 290
Linking Number; Seifert Forms and Matrices......Page 293
From Seifert form to Alexander Polynomial and Signatures......Page 300
Tristram-Levine Signature......Page 303
Potential Function and Tristram-Levine Signature......Page 306
A Combinatorial Formula for the Signature of Alternating Diagrams; Quasi-alternating Links......Page 311
Quasi-alternating Links......Page 318
References......Page 320
Generalizing Property R......Page 325
Property 2R......Page 327
The 4-manifold Viewpoint: A Non-standard Handle Structure on S4......Page 330
References......Page 333
Introduction......Page 334
Topological Enzymology......Page 336
Site-Specific Recombination......Page 341
Knots and Tangles......Page 344
The Tangle Model for Site-Specific Recombination......Page 349
The Topology of Tn3 Resolvase......Page 353
References......Page 357
Workshop Talks......Page 361