On Knots. (AM-115), Volume 115 9781400882137

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Annals of Mathematics Studies Number 115

ON KNO TS BY

LOUIS H. KAUFFMAN

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JERSEY

1987

Copyright © 1987 by Princeton University Press ALL RIGHTS RESERVED

The Annals of M athem atics Studies are edited by W illiam Browder, Robert P. Langlands, John M ilnor, and Elias M. Stein Corresponding editors: Stefan H ildebrandt, H. Blaine Lawson, Louis Nirenberg, and David Vogan

Clothbound editions of Princeton University Press books are printed on acid-free paper, and binding m aterials are chosen for strength and durability. Pa­ perbacks, while satisfactory for personal collections, are not usually suitable for library rebinding.

ISBN 0-691-08434-3 (cloth) ISBN 0-691-08435-1 (paper)

Printed in the United States of America by Princeton University Press, 41 W illiam Street Princeton, New Jersey

☆

Library of Congress Cataloging in Publication data will be found on the last printed page of this book

To

the M e m o r y A ndres

Re y e s

of

CONTENTS

P R E F A C E .....................................................ix I.

I N T R O D U C T I O N .............................................

3

II.

L I N K I N G N U M B E R S A N D R E I D E M E I S T E R M O V E S .............

9

III. IV. V. VI.

THE C O NW A Y

E X A M P L E S A N D S K E I N T H E O R Y ............................... 42 D E T E C T I N G S L I C E S A N D R I BB O NS ,

13. 14. 15. 16. 17. 18. 19. 20.

VIII.

AF I R S T

PASS.

70

Q u a t e r n i o n s a n d B el t T r i c k ....................... 93 R o p e T r i c k ........................................... 98 T o p o l o g i c a l S c r i p t .............................. 100 C a l c u l i ............................................... 103 I n f i n i t e F o r m s ..................................... 106 Q u a n d l e s .................... 110 T o p o l o g y of D N A ..................................... 113 K n o t s Ar e D e c o r a t e d F i b o n a c c i T r e e s ............ 115 A l h a m b r a M o s a i c ..................................... 1 2 0 O d d K n o t ............................................ 121 P i l a r ' s F a m i l y T r e e ................................ 122 Th e U n t w i s t e d D o u b l e of the D o u b l e of the F i g u r e E i g h t K n o t ............................. 123 A p p l i e d S c r i p t — A R i b b o n S u r f a c e ............. 124 K i r k h o f f ’s M a t r i x T r e e T h e o r e m .. ............... 129 S t a t e s a n d T r a i l s ...................................132 T he Ma p T h e o r e m .....................................147 The M o b i u s B a n d .....................................152 Th e G e n e r a l i z e d P o l y n o m i a l .................... 155 T h e G e n e r a l i z e d P o l y n o m i a l an d R e g u l a r I s o t o p y .............................................. 163 T w i s t e d B a n d s ........................................179

SPANNING SURFACES AND SEIFERTPAIRING

..............

181

R I B B O N S A N D S L I C E S ....................................... 208

IX.

ALEXANDER

P OLY N OM I AL AND B R A NC H EDC O VE R I N G S

X.

ALEXANDER

P O L Y N O M I A L A N D AR F

XI.

. . .

M I S C E L L A N Y ...................................................92 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

VII.

P O L Y N O M I A L .................................... 19

.

. . 229

I N V A R I A N T ...............252

F R E E D I F F E R E N T I A L C A L C U L U S ..............................262

vi i

XI1 .

. . . . . . . . . . . .

271

. . . . . . . . . . . . . . .

299

. . . . .

327

CYCLIC BRANCHED COVERINGS

XI11 . XIV . XV . XVI . XVII . XVIII . XIX .

SIGNATURE THEOREMS

G-SIGNATURE THEOREM FOR FOUR-MANIFOLDS

SIGNATURE OF CYCLIC BRANCHED COVERINGS . . . . . 332 SLICEKNOTS .

337

. . . . . . . . . . . . . . . . . .

345

CALCULATING or FOR GENERALIZED STEVEDORE'S . . . KNOTS . . . . . . . . . . . . . . . . . . . . . . 355

. .

366

. . . .

417

. . . . . . .

444

SINGULARITIES. KNOTS AND BRIESKORN VARIETIES

APPENDIX: TABLES:

. . . . . . . . . . .

AN INVARIANT FOR COVERINGS

Generalized Polynomials and a States Model for the Jones Polynomial . . . .

Knot Tables and the

REFERENCES

L-Polynomial

. . . . . . . . . . . . . . . . . . . . . . .

474

PREFACE

These notes panded

version

on

the

th e or y

of a s e mi n a r

G e o m e t r i a y T o p o l o g i a at Z a r a go z a,

Spain during

of k n o t s

held

in

the U n i v e r s i d a d

the w i n t e r

of

the

this a u t h o r

(we b e l i e v e ! )

careful

Du e

of

and

to

the

the m e m b e r s

the e n e r g y

set of notes,

de

de Z a r a go z a,

1984.

e n t h u s i a s m a nd p e r s i s t e n c e was g i v e n

ex­

the D e p a r t m e n t o

supernatural seminar,

c o m p r i s e an

of

to r e c o r d a

to r e l i s h

the

process. The n o t e s d i a g r a m moves,

begin with an d

( pe r h a p s

steep ly ) ,

problems

of k n o t

the mo s t

elementary

li n k i n g numb er s . using minimal

c o b o r d i s m an d

Then

they m o v e

t e c h n ic a l

the Arf

concepts

quickly

apparatus,

invariant

of

to

(Chapters

1 t h r o u g h 5). Chapter

6 is a m i s c e l l a n y ,

course.

It c o n t a i n s

ideas,

The

sections

this

last

the a u t h o r ' s polynomial

of

geometric

( [ HO MFLY],

This polynomial izes

the c l a s s i c a l

s how h o w i sotopy

compiled

si d et r ip s ,

chapter

musings [J01],

on

the

[J02],

new polynomial

of k n otted,

can be u s e d

s p e c ia l

topics. of

first g e n e r a l i z e d

[J03]). invariant

that g e n e r a l ­

an d C o n w a y p o l y n o m i a l s .

the g e n e r a l i z e d p o l y n o m i a l invariant

an d

the

c o n t a i n an e x p o s i t i o n

is a p o w e r f u l

Alexander

throughout

a rises

We

as an a m b i e n t

t w i s t e d bands,

an d h o w

to d i s t i n g u i s h m a n y k n o t s

the from

X

PREFACE

their

m irror

images.

Chapters geometric covering

7

kn o t

through

18

then d e v e l o p m o r e

t heo ry— w ith co ver ing

spaces.

(combinatorially)

polynomial

and

k no t

to s k e i n

t h eo r y b e g i n s

spanning

surfaces

S-equivalence S-equivalent

Seifert ^0')

7.

It

to p r o v e

that

the k n o t

ends

w i t h a ke y

identity

(v^

exe r ci s e, is

on

to s ho w

image by

points

to

asking

We d i s c u s s

isotopic knots the p o t e n t i a l

the S e i f e r t

have function

signature

the

The

in C h a p t e r

changes

image.

r e a de r

for

p a i r in g .

introduced

its m i r r o r

i aK (A -1) .

Th i s

is

* s not: a m P h i c h e i ral .

^42

that

mirror

to

the

i n t r o d u c t i o n of

sign

Chapter

to p r o v e

7

the

the C o n w a y p o l y n o m i a l )

^42

the k n o t

of

into g e o m e t r i c

p r o d u c i n g a m od e l

is a ls o

is r e p l a c e d by

—— i v^(2>l-1 )/| Vj^(2>l-1) | = that

terms K

when

Then

i n tr o du c ed , in

the

p a ir i ng .

that a m b i e n t

of a k n o t

is ea s y

7 with

p a i ri n gs . is

(Alexander)

The a s c e n t

the S e i f e r t

the C o n w a y p o l y n o m i a l signature

the C o n w a y

theory.

and prove

= Det(t0-t

to

in C h a p t e r and

s pa c es a n d b r a n c h e d

6 the r e a de r has a l r e a d y b e e n

By C h a p t e r

introduced

te c h n i c a l

c an n o t

then u s e d The

the o p e n q u e s t i o n

the p o s s i b i l i t i e s

exercise

be d i s t i n g u i s h e d

the g e n e r a l i z e d p o l y n o m i a l . of

i n he r e n t

to show

fro m This

g oe s

its exercise

s e t t l i n g a m p h i c h e i r a l i t y and in n e w

i nvariants

s u ch as

the

generalized polynomial. Chapter

8 returns

relationship with

to kn o t

the S e i f e r t

c o b o r d i s m an d d i s c u s s e s p a i r i n g an d w i t h

surgery

the

xi

PREFACE

curves

on

the

Alexander infinite cuss

spanning

polynomial cyclic

Seifert's

cyclic

terms

covering

one

is a b ri e f

introduction

complement.

We

its

relation

of

groups.

is a n o t h e r

into a c l a s s i c a l

a nd

on h y p e r b o l i c

the w o r k

complements

of R i l e y

(via r e p r e s e n t a t i o n s

i n d i c a t e a d i r e c t i o n here. the

link b e t w e e n

t acular

results

Chapter and

shows

how

tion p a i r i n g

12

these n o t e s

to c o m p u t e

explanation

S ei fert

p a i r i n g a nd b r a n c h e d of

a key

both

for

the

st u dy

entrance

into

signature the

for

exe r ci s e, by D e R a h m , on kn o t We

o nly

exercise

w ork and

T hi s the

coveri n g.

signatures

of

of

repre­

the

is

spec­

school.

on a s s o c i a t e d m an i f o l d s .

with a computation

this

their h o m o l o g y v i a

(more m o d e r n )

affine

key

11

calculus

g r o up s ).

cyclic branched

another

example

an d R i l e y ' s

of T h u r s t o n an d his introduces

Chapter

paper

Nevertheless,

in

Theorem

structures

of kn o t

of

the A l e x a n d e r

eight).

p r o vi d in g an entry point to

dis­

invariant

free d i f f e r e n t i a l

T his

the

the

Levine's

It ends w i t h an e x e r c i s e a b o u t

of kn o t

to

the Arf

the v a l u e

the

to

computing homology

(ta ke n m o d u l o to

the

and

an d p r o v e s

to

9 relates s pa c es

10 d i s c u s s e s

invariant

at m i n u s

for

an d

pai r in g ,

polynomial

sentations

the kn o t

method

Chapter

the Arf

of R a l p h Fox.

of

c o v er i ng s ,

the S e i f e r t

relating

to b r a n c h e d

original

p a ir i ng .

of

Chapter

covering

(b r an c h e d )

Seifert

surfaces.

coverings the

intersec­

leads

to

relationship The

chapter

torus knots.

c omputations,

topology

again,

an d

of

ends T h is

is

for an

of a l g e b r a i c

xii

PREFACE

singularities

( w h i c h we d i s c u s s

Chapter

introduce

12 we

m a n i f o l d an d knots

and

covering of

sh o w h o w

links are sp a c e s

the

of

signatures

in fact

19).

the

an d

signatures

Within

signature

of a

w-signatures

of b r a n c h e d

eigenspaces

in

of

cyclic the case

(J-s i g n a t u r e s ) .

giving

the

13 we p r o v e

signature

the p ieces.

manifolds that ants. on

the c o n c e p t

(or of a p p r o p r i a t e

In C h a p t e r

of

in C h a p t e r

signatures

to d i s c u s s

s ur f aces, basic

(case

signature

an d a n a r g u m e n t

gives this

of

of

skew

then a s s e m b l e

In C h a p t e r

the

a nd

the p r o d u c t

the case

defects

two m a n i f o l d s

that

signatures

these g e n e r a l

g-signatures

in

and

a nd use

of k n o t s

We p r o v e

g-signatures

of a u n i o n of

We a l s o p r o v e

va n is h ,

eigenspaces

the N o v i k o v A d d i t i o n T h e o r em ,

for

14 we b u i l d

links are theorem

for

these

to show

concordance

of

invari­

si g n a t u r e s , signatures

c y c l i c a c t i on s . forms)

of b o u n d i n g

results

for

in terms

in terms

We

cyclic

results

a n d go of

compute

actions

to o b t a i n

on the

four-manifolds. on

the

of C a m e r o n G o r d o n

results

of C h a p t e r

to g i v e a p r o o f - s k e t c h

G-signature

theorem

for 4 - m a n i f o l d s .

a complete

picture

of

through

18 are an e x p o s i t i o n

the

13

Th i s

G-signature

of

exposition

theorem

in

dimension. Chapters

15

of C a s s o n an d G o r d o n the p r e v i o u s covering

work,

spa c es

and

on

slice knots.

particularly upon G-signature

T hi s the

theorem.

of

the w o r k

depends

u p o n all

cyclic branched

xi i i

PREFACE

In C h a p t e r

19 we g i v e a n

of a l g e b r a i c

singularities

of k n o t s

links.

their

an d

specific space,

discussions

and

the M i l n o r

ideas.)

With

results

intimately

example

of

In p a r t i c u l a r ,

related

(large

generalized is g i v e n

relation

to

We

s p h er e s

conclude

of

the a u t h o r ' s

M u r a s u g i , and mo d el

to

the J ones

find to see

branched

chapter

in the

list

cov­

w i t h an

twos). further

developments a descrip­

polynomial

the J on e s

polynomial).

and

polynomial.

that

the author,

to M o r w e n T h i s t 1e t h w a i t e — all for

covers

con­

via Bri eskorn m a n i ­

to a l t e r n a t i n g k n o t s (work due

are g e o m e t ­

is p o s s i b l e

two-variable for

sus­

fro m a l g e b r a i c

In p a r t i c u l a r ,

st at e s mod e l

conjectures

latter

this

a s k e t c h of

polynomials.

to his

of

then d i s ­

the c y c l i c

the c l a s s i c a l

odd n u m b e r

contains

applications

century-old

sta te s

exotic

We g i v e

the p r o d u c t

it

a nd

t h ree-

We

cyclic branched

the e i g h t - f o l d p e r i o d i c i t y

The a p p e n d i x

We d i s c u s s

(The

t he o ry d e r i v e d

of Se i fert.

2 ( k ,2 ,2 , • • • , 2 )

about

space.

the or y

discus­

of e x a mp l es .

s u s p e n s i o n a nd

of

This

the e m p t y knots,

about

setting.

ering m et ho ds

its

in kno t

topology the

as p r o j e c t i v e

co nstructions.

the c o n s t r u c t i o n s

folds ar e

c ov e r i n g s .

examination

fibrat i on ,

the

relation with

dodecahedral

the c y c l i c

struction many their p r o p e r

as

to

Brieskorn varieties

2 (2 ,2 ,2 )

of

2(2,3,5)

ric c o n s t r u c t i o n s

tion

its

We d i s c u s s

by d e t a i l e d

pension and product

h ow

an d

relationship with branched

sion p r o c e e d s

cuss

introduction

using

s et t le to K u n i o this

xi v

PREFACE

Knot

t h eo r y

topology.

It

a d e ep u n d e r p i n n i n g

is a b e a u t i f u l

ramifications these p a g e s

comprises

that

spread

reflect

With great

this

s u bj e c t

in its

throughout

all

C a r m e n Sa font,

Esteban

I n d u r i a n an d E l e n a M a r t i n

for

through

tangled

thanks

Vaughan

Jones,

Pila r

terrain.

covering

Special

an d

J o a n B irman,

M a r i o R as e t t i ,

Sostenes

c on ver sations.

Special

Ra ndall

Cameron Gordon the

texts

knots Ms.

into

Illinois of

the

f irst

draft,

the U n i v e r s i t y producing

the

of

an d

Iowa

final

drawing

of an

Ra y L i c k o r i s h , Dennis

Massimo Agnes

to my

infinite

Ro s em a n,

Ferri,

for h e l p f u l

stude n ts ,

Ivan

an d m a t h e m a t i c s ,

and

to all

on

my

to D a l e R o l f s e n

i n t r o d u c i n g me

tale.

the

I am g r e a t l y

g-signature

of C h a p t e r s

thanks

at

for

text.

14

18.

last

typing

mathematics

of

this

of job

typist

collaboration

st ages

s lice

To

the U n i v e r s i t y

extraordinary The

to

to

to w e a v e

theorem and

for an e x c e l l e n t

to A d a Burns,

those

indebted

in a l l o w i n g me

Head Math Typist

at C h i c a go ,

journey

for

the e x p o s i t i o n s

S h i r l e y Roper,

this

Joseph Staley

tangled

work

Rodes,

listening and

for his g e n e r o s i t y

of his

Maite

for

of n a t u r e love a

May

Winker

to n e w

that

sharing

Jon Simon,

al s o

and Steve

to F r e d e r i c k

f ri e n d s

his

Alvaro

H u g h Morton,

thanks

contributing, realms

Valle,

Lins,and Corrado

Weiss,

with

of g e o m e t r y .

to K e n M il l et t ,

Larry Siebenmann,

K e i t h W o l c o tt ,

H an d le r ,

del

to r e p r o d u c e

space,

right,

I thank Jose M o n t e s i n o s ,

L ozano,

cyclic

own

spirit!

pleasure

for k i n d p e r m i s s i o n

of g e o m e t r i c

at

in

project

PREFACE

we r e p a r t i a l l y and of

s u p p o r t e d by O N R G r a n t No.

the S t e r e o c h e m i c a l Iowa,

xv

Iow a City,

Topology

Project

N0014-84-K-0099

at

the U n i v e r s i t y

Iowa.

C hi c ag o ,

F e b r u a r y 1986 an d Iowa City, D e c e m b e r 1986

On Knots

I INTRODUCTION

These notes

constitute

k no t

t he o ry

that

is

Trip

[FI].

We

ly,

sometimes

Y,

also

st ru c tu r e,

is kn o t

studies

the p l a c e m e n t X

and

is Y

S

3

S

is E u c l i d e a n

then we h a v e

X

1

or

Y

kn o t

the e m b e d d i n g s

of

2

the

three-sphere

of

IR3 .

S

3

2

is

a "knot

the one below.

that knot

s pa c es

Y.

Here

often means

(isotopy,

for

X

an d

the ho w

up

to

exam p le ) .

2 2 = 1, x a n d y real} = { ( x , y ) | x +y

The

theory. S

= { ( x , y , z , w ) | x +y +z +w

like

sometimes

three-space

classical

course,

in

is

Given

classify

= { (x ,y ,z ) |x ,y ,z real

2

Of

One a n s w e r

problem-'

a nd

of

the c i r c l e

IR

studies

occasional­

w ith p h i l o s o p h y

m a y be p l a c e d w i t h i n

form of m o v e m e n t X

to

of F o x ’s Q u i c k

to d i g r e s s

sometimes

t he o r y a b o u t ?

is u s u a l l y an e m b e d d i n g ,

If

free

sp ir i t

in the d i r e c t i o n of a p p l i c a t i o n s ,

classify how

some

feel

in the

introduction

ideas.

W ha t th eo r y

leisurely

(we h o p e! )

shall

w i t h an a n a l o g o u s general

a

2

1

in

n u mb e r s } , Classical

3 IR

or

= 1; x , y , z , w

the o n e - p o i n t

on a rope"

ma y

corresponding

3

knot

th eo r y

in real}.

Note

that

compactification

follow a pattern c las sical

kn o t

is

4

CHAPTER I

o b t a i n e d by

splicing

introducing new

the ends

of

the

rope

to g e t h e r

tangling):

T

Once

the ends

kn o t tednes s .

( w i th o ut

are

spliced

The u n k n o t

t o g et h er U

it

( tr e fo i l)

is p o s s i b l e

to d e f i n e

is r e p r e s e n t e d by

U a n d a kn o t

is

said

to be u n k n o 1 1 ed

if

it can be d e f o r me d ,

CHAPTER I

without

Thus

tearing

the rope,

the

fo r m

The

trefoil

knot te d .

W

Th i s

above T

proving knottedness,

an d

n ess

to

that g i v e s There

an d we

are

shall

(I c a l l e d p ar t is

sent

knots.

different

use m o r e

a p p r o a c h will

here

rise

be of

it

turns

into

the u nknot.

is u n k n o t t e d .

(two

requires

unt i l

5

sketches

proof!

A nd

a b ov e ) it

is

of c l a s s i f y i n g the n e e d way s

for a

is a c t u a l l y the q u e s t i o n

types

to a p p r o a c h

s uc h a The

c o m b i n a t o r ial a n d p i c t o r i a l

to c l o s e l y a b s t r a c t

K not the

Theory

theory,

first kn o t

in [Kl].)

rope d r a w i n g s

becomes

of k n o t t e d -

t h e o r y of knots.

than one ap p r o a c h .

it Fo rm a l

of

that

theory. The

idea

repre­

CHAPTER I

6

We

call

the p i c t u r e

contains

all

k no t

of

out

embedding y o u mu s t

rope a n d

understand

of

the c u r v e

Thus

we

st ar t

or b e l o w

rig ht a k n o t information

it p r e s e n t s

of a circl e ,

lift at

the

the n e c e s s a r y

p ar t

and

on

S

1

,

IR

that a b r o k e n

undercrosses

with a planar

the c r o s s i n g s

the p l a n e at

to

these

for

line

To

see

fo r m this

in dicates

the o t h e r

It

constructing

a specific 3

in

diagram.

the

for an embedding where

one

part.

graph

for m a c u r v e crossings:

w h i ch dips

above

CHAPTER

There

and n

are

t wo

hence

2n

at

potential

each

crossing:

knots

for

each

planar

graph

with

cros s i n g s . The

these

theory

notion

of

topology

can

then

And

we

spaces)

of

history,

and

Many

3

as

to

these will

in the

we

explain

this

way

theory

apparatus

with spaces. be

used

possible.

of

spaces Z,

and This

later

in

is

a:S

complementary

generalizations are

once

embedding

construct

associated of

begin

the the

also

topology

— » S

knot

apply

can

We

approach

t he

the

1

commences

diagrams. Another

S

choices

7

I

1

how the

to

S

algebraic (such work

as with

3 3

section.

the

and -a(S

1

abstract

to

study

) = Z. to

branched

covering

the

algebraic

also

in

notes.

of

the

initial

Two

are

particularly

Z.

has

a

placement

long

problem

worth

men-

tio n i n g . a)

S n — > S n + ^,

the

study

of

the

We

topology

approach these

deform

next

t ake

— » S

space

to

embeddings

of

an

CHAPTER

8

n - d i m e n s i o n a 1 sphere s i on s b)

I

into a s p h e r e

of

two d i m e n ­

higher.

W n — > S n + ^, being

same as

embedded

(a),

but

we a l l o w

to be an a r b i t r a r y

the

s p ac e

manifold.

3

One A

c an a l s o

link

in

S

look at 3

is a n

1 inks

in

S

embedding

and

in h i g h e r

of a c o l l e c t i o n

spheres.

of

c i r c le s .

Thu s

is

the

simplest

fascinating clude Rings, its

this a

example

phenomenon introduction

link

components

that ar e

of a of

li n ke d

link.

Linking

three-dimensional

with a picture

exhibits unlinked.

a

tr i a d i c It

is

of

space.

We

con­

the B o r r o m e a n

r e l a ti o n:

itself

is a

any

linked.

two

of

II LINKING

Ou r IR

f irst

NUMBERS

model

for

AND REIDE MEI STE R

the

theory

of k n o t s

We

that

is c o m b i n a t o r i a l ly based.

diagrams sequence K'.

K

and

K'

Equivalence

is a c o n t i n u o u s the other.

is d e n o t e d

diagrams

by

(see the

if

and

two k n o t

there

[R 1])

links

changing

sy m bo l

as

Reidemeister isotopy

of R e i d e m e i s t e r

through

proved

the

embeddings

moves:

1.

Reidemeister

9

Moves

There

a re

link

a K

in

into K ~ K'.

that

f r om

there

one

converse— making

iden t ic a l.

in

or

exists

are am b i en t iso top i c . m e a n i n g

deformation

lence a n d a m b i e n t

say

e q u i v a 1ent

of Re i deme i s ter m o v e s

Equivalent

types

ar e

MOVES

three

to

equiva­ basic

10

CHAPTER

It locally o t h er in

is u n d e r s t o o d on

the kn o t

strands

the moves.

is no t a that

o rd e r

pedantry,

Example:

but

is a l s o

locally

We

move

of

type

is a " h i g h e r

I i nv i t e

moves.

It

a re

o th e r

to b e p e r f o r m e d

understood than

that

no

those d e p i c t e d

Thus

example

And

these m o v e s

d i a g r am .

present

legitimate

this

agre e !

a re

that

II

him/her

i n sist

here

to s i m p l i f y

the

1.

order"

to

The

reader

move

formulate

of a

may

type

t he o r y

on p u r e

m oves

not

out

theory

(read

on!).

feel 1.

I

of h i g h e r of

CHAPTER

It

is a l s o

callv

understood

eauivalent"

that

ar e

II

11

two d i a g r a m s

eauivalent.

that ar e

"topologi-

T h us

an d

T hi s

part

moves. IR

of

We

that

get

thr o ws

K ~ 0 the

equivalence

really mean

E x e r c i se 2 . 1 . that

the

beyond

the R e i d e m e i s t e r

there

exists

a homeomorphism

one u n d e r l y i n g g r a p h

Give

an

ex ample

(the u n k n o t )

equivalence

E xe r c i se 2.2.

that

go e s

Prove

a nd

to

of a k n o t K

of

the other.

diagram

requires

a

type

K

such

3.

move

to

started.

that

the

following

process

will

always

CHAPTER

12

produce

an u n k n o t t e d

encounter start,

a previously

(------ )

Call

line,

dr a w i n g .

undercross

walking along

Whenever it.

we

crossings

say

more

than one

component

of a u n i v e r s e

link of

4-valent

is a

yo u

Return

to

are

vertices

trefoil

component

I mean a curve

the g r a p h a n d a l w a y s

components or

by

graph with

Th u s

may have

Here

Also,

a planar

a uni ver s e .

Such a graph

a knot

drawn

Start

eventually,

E x e r c i se 2 . 3 .

The

d i a gr a m:

II

as

un i v e r s e . in

obtained

c r o ss ing at

by

a cros s i n g .

the p o t e n t i a l

components

of

the p r o j e c t i o n .

that a k n o t

alternate

under

or —

link over

is a l t e r n a t i n g -- u n d e r

—

if

over

its -

•••

as

CHAPTER

you

traverse

any

component.

13

II

is

Thus

alternat-

ing . Prove:

Any

k n o t or

1 in k .

We the

now

universe

consider

components,

and

is

the

projection

of

2-componen t 1i nks . we

orient

them,

then

an

If we

alternating

a

and

wish

are

P

to

define

a

3 linking

number

Ik(a,|3)

=

lk(L)

(where

L

= a U

j3 C

S

)

so

tha t

This to

do

will this

conform we

with

associate

the a

usual s ig n

r i g h t - h a n d - r u 1e . e

to

each

In

crossing.

order

CHAPTER

14

DEFINITION. Let

a fl P

Let denote

L = a U P

II

be a link of

the set of crossings

two components.

of

a

with

(5.

Then

This

formula defines

the

linking

number

for a g i v e n d i a g r a m

Exam pIe :

lk(a.|3)

=|

E x a m p Ie :

lk(o.jB)

=

Example :

lk(a,/3)

=+2.

E x a m p Ie :

lk(a,p)

=0.

\

( 1 + 1)

=

1.

(1-1) = 0.

CHAPTER

This

last

l i nk e d

example

even when

is J.H.C. their

15

II

Whitehead's

linking number

link.

Links

can be

is zero.

[ A n o t h e r v e r s i o n of the W h i t e h e a d link]

E x e r c i se 2 . 4 .

THEOREM.

of a m b i e n t

lk(L)

last

and

L'

= lk(L').

exercise

of

links

really

Borromean

L

oriented

Linking

number

2-component is an

invariant

Istopy.

invariant ar e

the

L ~ L',

If

l i n k s , then

This

Prove

links

r i ng s

ar e

Let

that

the

at

and gives

linked!

E x e r c i se 2 . 5 . differ

shows

not

L site

that us

But yet

a nd of

lk

our

is a

f irst

topological

proof

the W h i t e h e a d

that

link a n d

some the

c ap t ur e d.

L one

be

two

two-component

cr o s s i n g ,

as

links

s h o w n below.

16

CHAPTER

G i v e n a kn o t

or

link

W,

II

define

1

if

W

has

one

component

0

if

W

has

more

is o b t a i n e d

from

C(W)

Suppose

that

W

than one L

or

component. L

by

splicing

out

the c r o s s i n g :

Show: lk(L)-lk(L)

(This

is a

to g i v e

new

triv ia l

Prove

tion p r e s e r v i n g

Mobius

=

but

the p a t t e r n w ill

generalize

invariants!)

E x e r c i se 2 . 6 .

h( M^ )

exercise,

= C(W)

t here

h o m e o m o r p h i sm

where bands,

that

and

respectively.

d oes

not

3 3 h :IR — » IR are

r i gh t

exist

an

s uc h and

orienta-

that

left

handed

17

CHAPTER II

Consider

(Hint: with

its

a

linking n umber

of

the

c ore

of

the b a n d

e d g e .)

E x e r c ise 2.7.

Link,

Twist,

W rithe.

N ot e :

and

(Isotopies Hence

we

ca n h a v e

where

twi s t ing

relative

situations

to

the e n d - p o i n t s )

w i t h a d o u b 1e - s t r a n d e d

link

18

CHAPTER

is e x c h a n g e d

For

for wr i th ing

an a p p r o p r i a t e

T(L)

= twist

can p r o v e

II

of

c la s s

L

an d

of 2 - c o m p o n e n t W(L)

= wr i the

lk (L 1 = T fL 1+ W ( L ) .

You

links of

L

L

define

so

that

you

should have

T

T

T hi s

fo r mula,

gists an d and

0

lk(L)

studying

[ B C W ] .) W

= T(L)+W(L),

c l osed,

You

can

by p l a y i n g

,

1 .

W

has

d o u b 1e - s t r a n d e d

see

the e x c h a n g e

w i t h a r u bb e r

E x e r c i se 2 . 8 .

Classify

the wa y s

wrapped

around

a c y li n d e r .

battery

problem

a small

transistor

[This

by B r a y t o n Gray. battery.)]

band

been used DNA.

(See

phenomenon or a

a rubber

by b i o l o ­ [ W H ] , [FB] between

telephone

band

T

cord.

can be

is c a l l e d

the

rubber

(He w r a p s

his

band

band,

around

Ill THE CONWAY

We n o w kn ots

and

variant [Cl], by

links.

of

It

a more is

[K2]).

This

1.

To

each

Alexander

v ^(z)

receive

identical

AXIOM

2.

If

K ~ 0

AXIOM

3.

Suppose one

K

Remarks.

or

€ Z[z].

invariant

three

c r o s s i n g as

K

Equivalent

(the u n k n o t )

that

lin k

of

oriented

(see

[Al]

and

is d e s c r i b e d

then

knots

there knots

The

r in g

or

v v = 1. K.

links

differ

at

the

s h o w n below:

(We call

Z[z]

and ,.

K

- Vrr = z v T . K. L

is a s s o c i ­

K ~ K'

polynomials:

K

then

polynomial

polynomial

o r i e n t e d k not

l inks

of

invariant

axi o ms :

ated a polynomial

site

powerful

the C o n w a y p o l v n o m i a l . a r e f i n e d

the c l a s s i c a l

[Kl],

three

AXIOM

introduce

POLYNOMIAL

is

L

this

the

19

the e x c h a n g e

rin g

i d e n t i tv . )

of p o l y n o m i a l s

in

z

20 with v K (z)

CHAPTER

integer

coefficients.

Thus

these a x i o m s

2

= a Q f K J + a ^ ( K ) z + a 2 (K)z +•••

n = 0

,

V v (z)

is a p o l y n o m i a l ,

z er o

III

1

a n (K) v / € Z

on a g i v e n As we

these

kn o t

shall

these

for

see,

invariants

is a n

are

n

except

where,

invariant i nt e g e r

assert

for

each

of

K.

that

Since

invariants

are

all

sufficiently

large.

for

a ^ , a^

and possib ly

What

they m e a n g e o m e t -

mysterious!

do

a^,

r ic a 1 l y ? The related

axioms to

exchange & Axiom a^,

the

do a s s e r t invariant

relation:

3).

We

a

shall

an d

to p r o v e

For

now,

that

each

a^f K)

invariant

that

we a s s u m e

.n (K) n+1 v 7

is

by a c o r r e s p o n d i n g

..(K)-a (K) = a (L) n+1 v 7 n+1 v 7 nv 7

use

a

this p r o p e r t y the a x i o m s

a re

consistency

to

(translate v

interpret

a^

an d

co nsistent.

and

set up

some

calcula-

t io n s .

LEMMA

3.1.

[ R ec a ll with

If

that

diagram

disjoint

a

L

link

is a split

is

containing

neighborhoods.

is a spl it

link.]

split

link

if

it

then

is e q u i v a l e n t

two n o n e m p t y Thus

= 0.

parts

that

to a live

link in

CHAPTER Proof.

If

two p a r t s We m a y below.

then K

vL

■

split

strands

K

Axiom

ar e 1.

21

then we m a y a s s u m e r e l a t e d as

fo r m a s s o c i a t e d and

^

For

is

with

VK = VK = °.

L

III

shown below

links

K

equivalent Therefore

its

and

via a 0 =

K

diagram on

the

as

shown

217-twist. - v^- =

has

right.

Hence ’

hence

example-

Re ma r k.

You may

enjoy

can be a c c o m p l i s h e d

Hint.

Prove

proving

the

lem ma

via Reidemeister

a generalization

of

type

that

moves.

2 move:

the

217-twist

22

CHAPTER

We us e For

the

fact

a proo f,

that

see

isotopic

varying

continuously

arising

small

the

embedding

The

lemm a

OOOO easy

and

to do

in

or

links

is a f a m i l y t

su c h

to a m b i e n t isotopy.

of

that

K,

K'

Kq

= K

that

the f a m i l y v a r i e s s mo o t h l y . context

neighborhood is

of a n y

IR

tame

point

= K'.

or S

3

are K nots

(in a

on

the k n o t

or

standardly unknotted). us

that all

receive

recursive

ar e

in

and

the e m b e d d i n g s

in our

ar e

embeddings

that

tells

,•••

Two knots

there

fr o m d i a g r a m s

sufficently link

if

we m a y a s s u m e

differentiable

is e q u i v a l e n t

[Rl].

ambient

Here

~

III

of

the v a l u e

the u n l i n k s : 0

from

v.

O O , It

o o o is n o w

calculations:

Examp 1e 3.2

vK = 1 + z2

CHAPTER Example

3.3:

& oS

L

He r e

L

lk(L)

= 0.

tion

is

L

the W h i t e h e a d We

similar

and h enc e we get

23

III

see

link

that

L ~

to E x a m p l e

3.1)

=

v^ -(-z)

1+ z

.

= z(l + z

from C h a p t e r an d

we h a v e

Putting

2

W

this

II.

thus

Thus (by c a l c u l a ­

Vj- = - z .

W ~ &

information

together,

).

3 •

Example

3.U:

•

v t

z

•

€0

We n o w h a v e

d en o t e

Thus

=

with

~

and

2n

W ~ U.

crossings.

Therefore

v y -z = z • 1

2 vT

= 2z.

2

>

%.

For

Let

e x ample,

L

24

CHAPTER

T he

same

reasoning

(by

III

induction)

sho ws

that

= nz. n

Since the

l k ( L n )= n = a ^ ( L n ),

coefficient

relation with

DEFINITION by

the

of

z n in

v^]

L

Let

we b e g i n

a^fK)

to g u e s s

is

a

be an y

knot

or

link.

C(L)

Define

formula fl = I [0

T hu s

C(L)

knots

a nd

links.

LEMMA

3.6.

Let

andz

respectively

if

L

has one

if

L

has m or e

is an

i n v a r i a n t of

a^

a^

and

a ^f K )

=

(ii)

& 1 (K)

flk(K) = {

(i)

Let

Then

3l q (K)-3Lq (K)

this

coefficient.

K,

and

it

denote

K

is

Since

e x a mp l e,

us e E x e r c i s e

t ran smute

to w h e t h e r (ii)

2.5.)

the

knots

when

an d

This

that

d i s tinguishes

coefficients

K

and

ha s

1

of

Then K,

links

two c o m p o n e n t s ,

o therwise.

= 0

switches

= C(K).

L

C(K) for all

switching.

according

than one c o m p o n e n t .

the C o n w a y p o l y n o m i a l .

in

(i)

Proof.

component,

( R e p e a t e d f rom E x e r c i s e

[ 0

a Q (K)

that

l i n k i n g n u mb e rs .

3.5.

C(L)

strand

[re c al l

L

the

says

be

statement that

there K

Again

a^

exists

of is

we one

let

see

that

or m o r e K,

K,

in A x i o m

the a x i o m invariant

a sequence

to an u n k n o t

2.2.), it has

r e l a t e d as

L

=1

components. be as

for under

of

or u n l i n k a^ ( K)

3.

(For or

0

Hence

in A x i o m

3.

25

CHAPTER III

Then

a^(K)-a^(K)

= C(L)

leave

the

the pr o o f

rest

This validity

of

s e c t i o n will of

foil.

3.7:

We w o r k

Watch

K

has

two c o m p o n e n t s .

as an e x e r c i s e

[Exercise

end w i t h a d i s c u s s i o n

our a x i o m s v i a an

coefficients.

Example

when

inductive

first w i t h m or e

the c a l c u l a t i o n

of

We

3.6].

the

definition

of

the

e xa m pl e s.

of

a^fK )

for

the

t re­

We h a v e

K

K

L

a 2 ( K ) - a 2 (K) = lk(L)

[Notation.: lk(L)

= 0

Let if

lk(L) L

= a ^(L)

does not

&

have

number ings

on

sense, that K.

= lk(L)

= a2 (U) = 0

a 2 (K)

= 1.

is o b t a i n e d

Thus

L

a 2 (K)

a 2 (K)

link.

two c o m p o n e n t s . ]

K a 2 ( K ) - a 2 (K)

this

or

3>

K

In

for a n y k n o t

computes from

= 1

a kind

of

links m a d e by

"self-1inking” splicing

cross­

26

CHAPTER

3.8:

Example

Note

that

Problem:

K

becomes

numbered

1,

need

some

for

2,

and

III

Calculate

unknotted

3.

if we

Obviously

notation

to h e l p

a^( K )

at

keep

for

s witch

this

crossings

point

there

is

the

tra c k

of

the

calcula-

r es u l t

of

switching

result

of

eliminating

t i on . (i)

(ii)

Let

S. ( K)

. th l

. crossing

Let

(iii)

denote

E ^ (K )

denote

i ^

crossing

Let

e^(K)

for a n y

of

denote

cr o s s ing of

Then

of

link

the

the

K.

the K,

the

the

by a splice.

sign

of

the

i

K.

k n ot

or

K

with

indexed

crossings,

W

K >-a n + l(S iK ) = - i ( K ) a n (E.K).

3 becomes

Axiom

CHAPTER

Using

this

notation

we h a v e

: (ej

= ^(K))

a 2 ( K ) - a 2 ( S 1K)

= tjlkfEjK)

a 2 ( S 1K ) - a 2 (S2 S 1K)

= e 2 l k ( E 2 S 1K)

a 2 ( S 2 S 1K ) - a 2 ( S 3 S 2 S 1K)

Since

S3S2S jK ~ 0 a 2 (K)

we

27

III

= &3 1k (E ^ S ^ ) .

conclude

that

= e 1 l k ( E 1K ) + e 2 l k ( E 2 S 1K ) + e 3 l k ( E 3 S 2 S 1K ) .

•

K

i

e i

=

-i

= +1 3 = _1

r lk(Xj) = 0 i

l k ( X 2 ) = +1

*• l k ( X 3 ) = 0 •

X 3 = E 3 S 2 S 1K

Note

that

this

calculation

is

sufficient

a 2 (K)

to c o n c l u d e

=

1

that

28

K

CHAPTER

is kn o t t e d . In g e n e r a l ,

signed

moves,

as a c e r t a i n it

is an

a "candidate"

a (K)

and

of

pa r e

that

invariance w i t h B all

restrict exactly

the

in

the

Take

a sum of

f o r m u l a above.

That

is,

(Denote universe

crossing Example’

end we

to h a v e

f r om a

the

that

first

time.

shall the k n o t

unknot

to

as

the kn o t

is not

--- •--- .)

over-crossing

(com­

to

in

f ol l o w i n g :

corresponding

of

defi

the p r o p e r t i e s

that u n k n o t standard

to

shall

that we

the d i r e c t i o n

the

then

is

as

for

and

in [BaM])

the b a s e - p o i n t

an

this

via

to def ine

this a p p r o a c h

it

create

try

to

on

in

To

invariant

The key

we do

a base-point

to

linking numbers

sequences

arise

is an

is n a t u r a l

to be p r o v e d a^fK).

that

lk(K)

i nv a ri a nt .

the u n i v e r s e

Choose

it

and Mehta

2.2.

how

sum of

switching

t hose

Exercise

and

as

we u n d e r s t a n d

the R e i d e m e i s t e r a 2 (K)

ca n be e x p r e s s e d as

a 2 (K)

Since

prove

a 2 (K)

linking numbers

CREATING

(a)

III

a cr o s s i n g . Walk along

the k n o t ' s each

K.

the

orientation

time y o u

cross

a

CHAPTER Let

KP

denote

operation. upon

t hose

The

K

spe cific

tation fro m

an d

the

n-i+1 this

and

diagram

label

in

this

depends

orientation. by

different.

the kn o t

by

1,2, ••• ,n

To

fo r m

its d i r e c t i o n

L ab e l

the

f irst

this of

orien­

crossing

set

the

in K

that

s e c o n d by

the

i ^

d iffer

n-1 ,

f rom

crossings

and ge nerally

new crossing

from

D

in

label

that

KP } by

is met in

t r averse. This

s uc h

that a re

the b a s e - p o i n t .

D = {crossings n,

is p r o d u c e d

unknotted

K P .We will

traverse

from

29

that

of b a s e - p o i n t

crossings

labelling

by

the u n k n o t

the c h o i c e

Compare

III

gives

that

sequence

S S • • • S 1K = K P n n-1 1

a s tandard standard

a switching

sequence

sequence

for

is u n k n o t t e d .

the o r i e n t e d

depends

upon

SgSjK

= Kp .

^ i ’^ 2 ’ * * * ’^n

K

knot

and

Call K.

the

this

The

choice

of

base-po i n t .

Example:

Each where

crossing it

is

is

f irst

labelled

(in

traversed.

the o r d e r

n, n - 1 , • • , 1 )

30

CHAPTER

DEFINITION

3.9.

S j , • • • ,S

be a s t a n d a r d u n h n o t t i n g

e. = e.(K) l lv J

of b a s e - p o i n t , to do

l

the

sh o w

oriented k n o t .

,S.

l -1

l -2

ci(K)

that

it

(i =

1

is

is a

sequence

• • -S.K

a(K)

formula

that

an d

this

be an

E.S.

X. l

by

We m u s t

o r de r

Let

and

a(K)

Define

K

III

v

=

K.

for

Let

1 , 2,•••,n).

J

e^lk(X.).

independent topological

a preliminary

Let

discussion

of

of

the

choice

i nv a r i a n t .

In

the u n k n o t

is r e q u i r e d .

UNKNOT

DISCUSSION

L et's verse

think about how

fr o m

p,

K

K

m a y look

I

I

I

an d

KP

dif f er .

As

Th e

crossing

encountered w ill

look

I

lab elled when

like

• * •

I

I

i

is

I

t

* * *

and

i

i

the

first u n d e r c r o s s ing p.

As

a result,

will

1a b e 1 s the

be

I ---I I* •••

I

changed

first

I to an

over-crossing

c ro s s ing c h a n g e

be t ween

in K

Kp . Let's

KP

this:

p I I is,

K.

I

t r a v e l l i n g f ro m

I I

That

tra­

like:

-------------------- i-------------------P

we

consider

the

>-•-- ----- K

p i

following

si t ua t i o n :

i

»-•---- :--- K

p 11

p

KP , and

CHAPTER

Here

the

s ite

of

fi r st the

crossing

fir s t

s t ances

that

crossing

i_s a split

III

31

occurs

after

change.

Under

p

is a l s o

the

the s e c i r c u m -

un 1 ink .

Example:

E^K^,

We

leave

the p r o o f

PROPOSITION a(K ) ,

Then of

base-point

point

Let

on

a.

To

suffice

emphasize

write

it ma y

K

s li d e

an

exercise.

be an o r i e n t e d h no t is

unlin k .

independent

diagram. of

the

choice

K.

through a crossing

we will or

this a s s e r t i o n as

as d e f i n e d ab ove,

It will

P roof.

of

3.10.

of

a spl i t

to show in

K

that

without

the p o s s i b l e

a ( K, p ). under.

Now This

we

can

changing

dependence

the b a s e - p o i n t leads

s l id e

to

two

the b a s e the v a l u e

on b a s e - p o i n t

may

s li d e

cases'-

over

32

CHAPTER III

Case

1.

--- »-•—

----

P

Here

K

and

KP

under

the c r o s s i n g

p

K

differ to

i

at

q

i.

i

crossings

are

the

and

Kq .

Thus

for

K

i.

and

And .

not

if

is a

then

a(K,p)

Case

K

and

KP

u n li n k.

and

the c r o s s i n g sequence

f rom

the u n k n o t

crossing.

The

KP .

s li d e

If we

labelled

for

I

proving

Case

discussion

l k ( E n K P ) = 0.

K

a nd

the

over-crossing

p

I

jl

'k

i

across

q

d o es

that

E KP n

Hence

--- »— •--------------

11

crossing

Since

I

1.

P

I1 I kp

line

just

Ik on

K

sequence

E S 1 - « * S 1K = E K p . n n-1 1 n

»— •-:--------

lies

between

= e ^ k f E ^ ^ • • • 'SjK) .

Therefore

2.

p

as

changing

we h av e

P

Here

all

is a stanc*ard

* S a stan-•---- -----

not

change

f rom

then

*

i Kq

prior

q

to K

the to

CHAPTER the

crossing

added If

one

the

d oe s

crossing

switching S

with S

n

between

the

set

sequence

for

,S

to

33

K

of

and

Kq .

changing

K t Kq

Thus

we h a v e

crossings

of

KP .

is

1 , • • • ,S. , fS.,S. , , • • • , S t n-1 l+l 1 l-l 1

switching

,S n n - 1

K,

change

III

the c r o s s i n g

-,,S. i+li-1

l abelled

i,

is a s t a n d a r d

1

then

sequence

for

KP . Thus

and

the

a (K,p)

f i r st are

the the

a n d all

i t^1

terms

i de n ti c al .

l k ( E ^ S j _ j • • #S ^ K ) . a(K,p)

i-1

the

crossing.

This

the

Then

term

remaining Note

in

for

a(K,q)

a(K,q)

has

the

term

is m i s s i n g

f ro m

the

sum

terms

that

sums

differ

by a s w i t c h at

suc h a s w i t c h

will

affect

linking numbers £ k + i l k (E k + is i+ k - r - s i+ is i - r ” s iK ) ~

k 'kP

e k + i l k (E k + is i+ k - r * * s i+ is is i - r " s iK > — only

for

i f the c r o s s ing

p o n e n t s o f the only

if

Note

that

that

the

(k+i)

l a b e 11ed

r e s u l t ing label s

And of

= e^ C ( E ^ X ) .

the d i f f e r e n c e s

.k ^

is a c r o s s ing of

a crossing

lk(X)-lk(S^X) sum of

1i n k .

i

k

this

is

true

we

yields'.

a (K ,p ) - cl (K ,q ) = - e i l k ( E . S i_ 1 * * - S 1K)

com­

if a n d

two c o m p o n e n t s

Therefore

two

+ e.A

of E ^ .

conclude

34

CHAPTER

where

A =

the

is a c r o s s i n g We al s o

sum of

know

unlink

for

first

return

just

indices A =

whose

lk(E

pletes

of a

backwards

k = 1,2,•••

t he

E.S

1 n

same

prior

link

2 and

Case

the p r o o f

2 of

unlink. through

it,

t he

switching

to

(i

S^S^S^K p

set

this

is

Since of

the the

link­

crossing

implies

= a(K,q).

that This

com­

the p r o p o s i t i o n .

Proposition

Here 1

of

k+i

is a

1

as b e f o r e

s u m of a n y

unlink

that

E^K.

the b a s e - p o i n t ) .

the

switchings

such

. • • • S . .S. n-1 l+l l-l

reasons

to

is

S

of

Hence a(K,p)

(for

is a s p l i t

from

that

S

Case

Example

e, . , k+i

two c o m p o n e n t s

split

ing n u m b e r

of

III

3.10):

~

.

(as b e l o w )

sequence.

Note

If w e

that

slide

then we will

E^Kq

q lose

CHAPTER

Here

S ^ K N ow

III

35

~ KP .

compare

the

computations

for

q

and

p.

a ( K , q ) ____________________ a ( K . p )

We

see

ut e d

clearly

across

Rema rk . fully, matic

how

the

If y ou you

will

properties

the

ot h e r

linking n umber terms

examine see of

in

the c a l c u l a t i o n

the p r o o f

of

that we a c t u a l l y the

linking

l k(E^K)

is d i s t r i b ­ of

Proposition used

number.

exactly Th a t

is,

a(K, p) .

3.10

care­

the a x i o ­ that

36

CHAPTER

lk(X)-lk(S.X) cal

= fc.(X)C(E.X),

invariance.

and used axiomatic

properties

a ^( K )

terms

in

way,

we

ance

for

This

is

create

of an

rather

of

the

for

a(K) using

of

creating

topologi­

corresponding

= a^(K), the

and

c a n be g e n e r a l i z e d

the

then we

c an d e f i n e

same a r g u m e n t .

definition

sequence

reminiscent

= 0,

this p r o o f

we p r o v e

inductive

like

is c e r t a i n l y

Once

a^

the w h o l e

it

{ >.

lk(OO)

As a result,

inductively!

III

an d p r o o f

coefficients

something

In of

this invari­

a ^ ,a ^ ,a ^ ,•••.

fro m n o t h i n g !

And

of

the V o n N e u m a n n p r o d u c t i o n

{{

} {{

ordinals

{{

}}.

{{

} {{

}>}.

}}

{{

} {{

}}}}•••■

E x e r c i se 3.11.

K

(a)

Work

out

a(K,q)

fr o m

(b)

Work

out

a(K,p)

an d

wi th (c)

this

d i ag r am .

compare

your

calculation

(a ).

Simplify

the d i a g r a m by R e i d e m e i s t e r

gram

K'

with

Find

v j( / •

fewer

crossings.

F ind

moves

to a d i a ­

a(K').

CHAPTER III E x e r c i se 3 . 1 1 a . amine

both

This

exercise

linking numbers

37

is d e s i g n e d

and

the p r o o f

to h e l p

of

you

reex­

Proposition

3.10. (i)

Let

L = a

U J3 be a

Let

l , 2 , # # , ,n

s uc h that

link of

be a set

two c o m p o n e n t s

of

S S . •••S. L n n-1 1

crossings

is a split

of

a ,f3. a

with

j3

link. S h o w that ----- -----

l k ( L ) = 6l ( L ) + e 2( L ) + - * - + e n ( L ) .

Example:

S ^L lk(L)

(ii)

Let

L = a U /3

Let 1 , 2 , •••,n that

be

= e 1(L) .

link of

a n y set is

= 2^_^e^(L)

C

(E^L)

some e x a m p l e s of u n k n o t

1on k n o t s a n d z er o

on

a ,(3.

in L

s uch

S h o w that -- — -----

C

as

defined

in

links.

diagrams

that a r e

not

u n k n o ts .

ha d

started

the d e f i n i t i o n

s ho w

link.

where

Give

as

crossings

a split

is

If y o u

two c o m p o n e n t s

of

3.5,

standard (iv)

be a

S S. • • • S . L n n-1 1

lk(L)

(iii)

= -1

split

that

i nvar i a n t ?

it was

with of

the

formula

l i n k i n g n u mb e r,

well-defined

an d a

in

(ii)

(or

(i))

h o w w o u l d yo u topological

38

CHAPTER

III

E x a m p Ie :

L

Switching opposite N ow

the

two

sign.

Therefore

onward

to

We h a v e

Re ma r k.

starred

(one

c omp onent).

just

as

easily

untangling

one

for

links.

the a v e r a g e

of

to

links;

a' value

it will

procedure

linking n um ber

as

of

a(K)

unlinks

(Define

by

this

o n e - h a 1 f of

is

w h at the

s ai d w o r k s sequence

independent

over

be d e n o t e d

is e x a c t l y

h av e

is a k n ot

we h a v e

Then

a.

K

a switching

summing

of

They

i nv a r i a n c e .

o nl y w h e n

Show

L.

= 0.

everything

component.

t ak i n g

this

the p r o o f

H ow e v e r ,

Now define

that

lk(L)

defined

point.

a

crossings

all

of

components

a'

is

the

a

as

befor e .

by

we do

sum of

w h e n we the

by baseand

extension Note

de fine

crossing

s i gns . )

Note.

a ^( L )

ca n be n o n z e r o

on a 3 - c o m p o n e n t

link: i s an

From now and

on,

link d i a g r a m s .

we a s s u m e

a(K)

is w e l l - d e f i n e d

examp 1 e .

on knot

CHAPTER

PROPOSITION d ia g r a m . (i)

3.12.

be an o r i e n t e d knot

is

invariant

links

If

K,

K

under L

and

the R e i d e m e i s t e r are

K

are

consistent

Th e

idea:

the c r o s s i n g s

p ar t

of

the

T hi s

moves.

a (K )- a (K ) =

lk(L).

L

satisfies

P roof. of

K

a(K)

axioms

link

r e l a t e d as

then

Thus

or

Then

a(K)

(ii)

K

Let

39

III

the a x i o m s and

ot(K)

Position i nv olved

switching

idea w o r k s

for

a^(K).

Hence

these

= a^( K ) .

the b a s e - p o i n t

so

that

in a g i v e n R e i d e m e i s t e r

"none" move

are

seque n ce . for m o v e s

of

type

1 an d

type

2 as

s h o w n below-'

Since the

a

is d e f i n e d

R-move

for

each

by a sum of

i nv a r i a n t s ,

we

form

sum w i t h o u t

changing

in

the

va lues. Th i s

idea a l m o s t

works

for

the

type

3 move:

can p e r f o r m the

CHAPTER

40

The

starred

switching above.) term

a

switched do

(*)

sequ e n c e . However,

or

in a n y

every

change

( Hence

this

it ma y And

equivalence,

by

may

or

a(K).

if

two

case,

term

be

still

the

spliced. spliced,

type

an y

proves

"none"

starred If

we want

looking

inv a r i a n c e .

m ay

be

we

can

at

a

still

simple

in

to do a nd

the

In a g i v e n

crossing

as

in

in q u o t e s

s w i tched,

we are

a ^ 1k (E ^ S ^ ^ •••S ^ K ) This

involved

no p r o b l e m !

two m o v e s

mo v e

be

we h a v e

presents

1k (E S _ ^ •••S ^ K ),

the move.

Thus

crossing

III

can be

hence

inherited

will

not

CHAPTER

Part

(ii)

Position

Hint:

the proof,

Exercise

will

the b a s e - p o i n t

(A small

osition 3.10-3.13)

-

s i gn

(This

so

it

Rewrite

inductively

Investigate

bx a

is a

See

+

[J01],

c as e

the

This

the

defines

of

consequences

the C o n w a y

s ig n !

t opo logical

is a s p e c i a l

n omia l .

3.13.

correctly!

project).

in A x i o m 3 for

(v^-Vjj- = zv^) polynomial

Exercise

completes

theory all

of

(Prop­ the

coefficients.

E x e r c i se 3 . 1 4 . the

left as

41

h

3.13

polynomial

be

III

Prove

invariant the

o f rep lac ing

Polynomial

that

the

resulting

of k n o t s

and

f irst g e n e r a l i z e d

[J 0 2 ] , [J03]

and

[ H O M F L Y ] .)

links.

poly­

IV EXAMPLES

Here 3. N o t e of

th e

and

we

continue

that

the

consistency

then

some

E x e r c i se

4. 1.

These

are

of

skein

v

For

z =

1,

Here

Fibonacci

n

with

= v T, K n

this

Chapter

the

knots

then

yields

component

THEORY

via

3 has

axioms.

theory

shows

SKEIN

calculating of

alternately

If

(along

end

AND

the

provided

First

(see

Conway

and

l i nk s.

axioms

some

that

Fibonacci

none

count

for

42

of

a proof recursions,

Thus

Series

these

the

Chapter

[Cl]).

v -v 0 = zv 1. n n-2 n-1

the

of

are

first

we

have

1,1,2,3,5,•••. equivalent

tw o).

CHAPTER

V1

V2 V3 v4 V5

V6

L

IV

43

44

CHAPTER

Thus

v K = V £ = ( 1 + z 2 )2 .

receive T he

the

proof

inequivalence

have

developed

c op y

of

to

the

this

show

so

far.

tr ef o il

example

that

(Why?)

same p o l y n o m i a l ,

of

K

later

and

K

ar e

K

not

subtler

that

K

image.

(Ex e rc i se :

Us e

and

K

equivalent.

methods

than we

is c o m p o s e d We

shall

Exercise

of a

return 3.15

to

inequivalent!)

4.2.

Find another

distinct

knots

that

wh ere

also

its m i r r o r

on.

knots

they ar e

requires

Exercise

P r o b 1e m :

The

but

No t e

and

IV

s h ar e

example

the

of a pa i r

of a p p a r e n t l y

same p o l y n o m i a l .

Given a polynomial

f(z),

fR e s e a r c h

investigate

K(f)

K ( f ) = ( K | v R = f }.]

SKEIN NOTATION When in

the

links

A,

B

an d

C

are

we

shall

©

and

C

write

0

A = B © C

v.

for

switching

ABC

an d

C

B = A 0 C.

a r e n o n a s s o c i a t i v e , an d

notation

as a

B

these

operator

= v.-z v ~. A C

T hu s

in

this

operations a conven­

[You m a y

t h i nk

of

form.]

v^ _vb = ZVC ‘ we

T he

they p r o v i d e

relationships.

By Axi om 3 we h a v e an d

related

form

A

ient

diagrammatically

could also

Hence define

vg0c = VB+ ZVC ©

and

0

45

CHAPTER IV in

Z[z]

tion, VA®B

and

by

f © g = f+zg

the

related

and

f 0 g = f-zg.

ideas a re

due

shall

venience. Theory

use

©

However,

as

B ~ B7

0

Conway

foll o ws :

and

an d

creates

Define

C ~ C'

primarily

Then

where

A', B7

fit t o g e t h e r

as above.

this

w a y we a l l o w

the p o s s i b i l i t y

m o r e than with

one

B 77

way!

and

compositions cally

C 77

to

B 7 ~ B 77 , skein

ate d ) .

and

Th u s

B7 © C7 C 7 ~ C 77 .

equivalence

fitting

be

say

if

these

the

C7

that

A ~ A 7,

are

we m a y

B 77

diagrams

compositions B 77 © C 77 -

this

in in

C 77 ~ C

T he

B 77 © C 77

equivalence

do

~ B,

m a y not

con­

Skein

whenever

together.

that

We w r i t e

calls

including equivalence

B7 © C7

we

he

and

there ma y be

also

B 77 © C"

equivalent.

e q u i v a 1ent

Thus

By

for n o t a t i o n a l

w ha t

A = B © C

that

let

to J o h n C o n wa y .

nota­

= V A ® V B'

We

and

This

resulting

be

topologi­

is a re

ske in defined

B7 © C7

relation

(and

so g e n e r ­

CHAPTER

46

More ©

and

are

generally,

9

ar e

topologically /.

Thus

(topological v

skein

links

Open

P r o b 1e m :

not

Is

just

a

f an c y

way

of

the

skein

n

components.

d e c o m p o s i t i on

classes

of

to

of An y

ab ove

then

equivalence

f r om

skein

the

links,

then

C#/sk

to

: % / s k — > Z[z].

skein-equivalent

knots

or

equivalent.

we ha d

Th i s of

the

saying out

knot

K = U 0

is a

skein

that,

on one

0 0

K

in

$

of

kn o t

a nd or

that

the

U^'s.

where

s k e i n - d e c o m p o s i tion

recursion, the

1 ist:

These

are

links.

link

(L © U)

A nd

& = '&/sk.

by

,•••.

knots

involving

L'

and

v

terms

the u n k n o t ?

00 of

of

L

of k n o t s

9).

two

involving

individual if

set

there a n o n t r i v i a l

,o if

©,

topologically

bottom

, 0 0

the

homomorphism

the g e n e r a t o r s

calculations

u = 0

L =

if

Clearly,

denotes

example

example

and into

their

(operations

skein-equivalent

U =

equivalent

%

F i n d an

the

compositions

equivalence)

that ar e

In

K

if

Z[z]

Exerc is e •

two a l g e b r a i c

equivalent.

is a w e l l - d e f i n e d

the

is

skein

IV

Let

all

of

this of

is

our

the g e n e r a t o r s =

00

can be w r i t t e n as

a

•••

0

s ke i n -

CHAPTER

IV

47

Example:

E = V

© L,

L = U2 0 U 1

E = Ul ©

(U2 0 U x )

t

v r = 1-z E E Note

that

we

have

We will simple

by

recall

that

denoted

is

now

the

sta t e

existence

Thu s

to

a

This

of

the sum

is d e f i n e d may

it

of

it

from

depend

skein of

by

this

the

is m a d e

decompositions. K

splicing

upon

trefoil.

proof

links

the

a nd two

choice

way:

very First

K 7,

strands of

K # K'

K' think

knot.

theorem whose

the c o n n e c ted

K I prefer

figure-eight

distinguished

K # K 7,

s h o w n below.

the

2

as

strands.

CHAPTER

48

THEOREM

4.3. V K#K'

(ii)

= V K V K'

K is

If

K,

Let

the

on all

K7

be

oriented knots

( p r o du c t

res u lt

strands

of

of

IV

in

links.

Z[z]).

reversing

K,

or

all

v

then

=

K

orientations

.

K

I

(iii)

Let

K*

by v

be

switching I ( z )

=

K ‘

(iv)

If

L

v k K

=

every

image

crossing

K

of

of

K).

(obtained Then

( “ z ) ■

is a

? L (-z)

lin k w i t h A

(-l)X+ 1^ L ( z ) •

components Hence

v

then , = (-1)X + 1 ?L .

E x e r c i se 4 . 3 .

Proof :

COROLLARY ponents, are

the m i r r o r

not

4.4.

If ^

and

L 0,

is a then

link w i t h an L

and

even number

its m i r r o r

i mage

equivalent.

E x a m p Ie :

L These

links

are

not

e q ui v a l e n t .

•

of

com-

i L"

CHAPTER

We

N ote:

have

(one

already

strand

shown

has

49

IV

that

orientation

reversed

fr o m

L)

No w

(figure

e ig h t

knot )

- V L +Z = z ( 1 - z ^ ) = z the v e r t i c a l

dotted

also! a xis.]

[This We

follows therefore

by

symmetry

have

the

around

following

tab 1e :

with any

component

o r i e n t a t i o n has

v = +z"

wit h any

component

orientation

v = -z

has

.

Therefore

the

50

CHAPTER

u n o r i e n t ed

Whi t e he a d

1 ink

W

IV

and

her

mirror

image

W'

inequivalent.

w

Example

U.5.

W e a v i ng

X, = RX. X,= R X

This

is

a

generates

simple the

chain

weave

is

stitch. of A

the

The

basic

form RA

recursion

that

are

Thus

X

. = RX . n+1 n

From

51

CHAPTER

IV

this

can c a l c u l a t e

RA

we

v

=

n

X

• n

RA

Splicing:

-zv.

LA LA

Thus X

v„A

=

RA

-z^VA. A

Vt

A

In our

case,

X

= R n X ri = R ( R n _ 1 X~) ~ R n - 1 X n 0 v 0J 0

n

is,

if yo u

becomes

switch

the

unknotted.

=

crossing

1-z

=

V^

= 11-z

V0

=

2

,

2

2 4J

You may

Vv

00

Then = X

00

X

,

then

it

n-1

.

2

6

4

be

t e m p t e d by

-z

would

In p l a c e .

6

V

2

there

= 1-z v Y

.

-z 4

1-z + Z + Z 1

RX

of

Tha t

1+Z2

= 1-z + z -z

ity! J

Hence

2

n V0

R n X~ . 0

R X 7 - X T ~ 0. 0 0

right-most

.'.

=

n

of

Whence

n

2

,

4

1-z + z - • • • ± z

=

2n + 2

8

this

be an X

’’i n f i n i t e =

n

(1+z

formalism

2

knot”

2

1-z V v , X n-1 )vY

=

1

to go

an d

to

X 00 .

we w o u l d

infin­ And h av e

52

CHAPTER

IV

1 2 a. = 1-z + z4

1+ z This

formal power

+ ** *± z^ n + ^. knot

X^??

But

series

is indeed

is there

the

limit of

such an entity as

One possibility

for

= 1-z

this

infinite

is a wild knot

form

2

in the

______

P The

Here such

’’wild point"

represents a homeomorphic that

hood of

the image has a wild point p

contains

the pair

(ball,

arc)

s tab i 1i tJ y:

3 IR

in

Every n e i g h b o r ­

Neighborhood (T K )

Nevertheless,

there

is a sense

in

from

K

and

does not

does have

the weaving

You can do any

K 00 ’,

then

finite amount

making him disappear.

3

in an unknot! K 00

In this case

3

— » IR

If you call

is a curious animal.

there

is an infinite

corresponding

that unknots

considerations

the

of unravelling of him without

composition of elementary moves : IR

in which switching

results

switched version

topological

1

RK CO = K 00 .

starred crossing

h

p.

S

(under a h o m e o m o r p h i s m ) to the standard

pair.

However,

morphism

image of

infinitely many crossings

(Neighborhood,

straighten out

this

p.

--

K

.

to a homeo-

In general,

for wild knots are more

the

CHAPTER

complicated. differently

with

Instead, as

in

infinitely

three

dots.

X 00

construe

the

infinite

knot

below:

many

But

we may

53

IV

tiei n g s

this

beast

in

the

doesn't

region live

indicated

by

the

in E u c l i d e a n

space! I suggest represented as

X^

the

by a p i c t u r e

= [X Q I X

quences

of

following

a s tab 1e equ iv a 1ence [a]

= [b]

if

some

k ,H .

then

RX 00 = X 00 . We

als.

Then

can use

knots

where

we

the

same

that

of

of

above; of

as de f ine

infinite

[ sLq ,3.^ , • • •] = [a]

infiniten-tuples.

= b n+a for all

if

n

stable

classes

se­

denotes T h a t is,

= 0,1,2,***

set R [ a Q , a ^ , * * * ]

X^

an d

= [ a ^ ,R a ^ ,R a ^ ,•••]

for

the p o l y n o m i ­

Thus VX

00

= [V X 'VX ’V X '* ’ *] Ao 1 2 =

a nd

to

Think

in a c a t e g o r y

cla s s

a n+k

out.

similar

fX 2 ,X3 ,•••]

ordinary

way

this

can be

r,, [1 + z

2

,

1,—Z2

4

- Z

, • • • ]

taken as a r e p r e s e n t a t i v e

,

of

the

formal

54

CHAPTER

power

stable

v.

00

(See

1 + z -z + ••*.

series Other

IV

infinite

sequences.

knots Thus

have

v ’s

that

only

exist

as

if

= [ 1 , - z ,z ^ ,- z ^ ,•••].

then

You m ight

say

vA

= (-1)°^° CO

[ K 3 ] .)

E x e r c i se 4 . 5 . weaves

Choose

and knots

Rema rk .

property sequence sequence

=

that

recursion

defined

R

on

the n

an d

sequences

( a ^ ,R a g ,R a ^ ,R a g , • • • ) .

lim R n-*» is

own

analyze

the

so ob t a i n e d .

We h a v e

R ( a Q , a j , •• •)

your

limit

(a)

of

R n (a)

= (a^.RaQ.R

invariant

under

R!

2

by

Th i s

makes

has

sense

3

a ^ ,R a ^ ,•••) Then

taking

the ni c e as and

the this

CHAPTER

stable

equivalence

(a 0 ,Ra 0 ,R 2 a o ’*'*) ~

classes

way

[a]

has the

( R a 0 ,R2 a 0 ,-.-)

~

k k+ 1 (R a ^ ,R a ^ ,•••),•••.

is a formal

55

IV

”R a^

that

~ (R 2 a 0 ,R3 a Q ,-- ) ~

In ot h e r

to say

effect

2

words

for

k

•••

[ a Q . R a ^ . R a^ , • • • ]

indefinitely

large.”

00

T hu s

R a^

is a g o o d n a m e

for

the

seq uence

[ a 0 ,Ra 0 ,R 2 a 0 , •••].

We

Re ma r k. Kw

could

denote

question

have

with

is,

what

o nl y

Generalize

with

Digression recording nected

some

k .6 .

That

4.6.

Let

then

proof

*

the C o n w a y

(say u s i n g

w e ’re

that is,

K K

~ 0 => K ~ 0

The

labelled

Reidemeister

While

a proof

unknotted,

the w i l d

knot

K^.

Let

s w it c h e d .

The

polynomial

equivalences

moves,

or

into

that

infinitely

an

use many

co n t r o l ) .

sum.

THEOREM

K # K'

many

with

V Koo^

for w i l d k n o t s

finitely

moves

crossing

is

Re search Problem. invariant

stayed

w ill

yo u

talking wild knots

and a nd

take

Ky

con­

to p r o v e

K ' be two kn ot s . are

form

If

K # K'

e a c h u n k n o t t e d . That

K ' ~ 0.

the

is w o r t h

c a n ’t c a n c e 1 k no t s u n d e r

we w i s h

a nd

it

of a d e tour:

is,

is

56

CHAPTER

Wild

Tame

Knots

So we h a v e

to

IV

Kno t s

— — — — — — — — — — — —

say

something

about

the

Tame

Knots

relationship

of

these

categori e s . A

(possibly

morphism kn o t s

a

a,p

orientation that

throws

1

3 — » S

of

— > S

3

we

preserving

words,

one

is

represented

the c i r c l e say

that

by an y

into

S if

a = p

homeomorphism

h

: S

3

3

.

homeoGiven

there — > S

two

is an

3

such

knot

h(a(S*)) to

= p(S^).

the oth e r

and

The

homeomorphism

transforms

the

h

surrounding

as we 11. One

without knot

: S

1

knot

p = h o a.

In o t h e r

spa c e

: S

wild)

says any

that,

standard

that a kn o t

wild up

points.

is

tame

Th u s

if

it mu s t

to h o m e o m o r p h i s m , looks

(ball,

arc)

pair.

it

is be

=

to a k n o t

equivalent

locally

like

to a

the

CHAPTER

IV

57

Standard

Pair

3

Fact:

Let

grams

K

T ha t the

is,

K,K' an d

K '.

for a d e t a i l e d

territory

Proof

of

be a Then

equivalence

topological

This

C S

fact and

equivalence discussion allows

return

T h e n we ma y

Cfl

us

=

of

moves

defined

this

only

if

is

by

K ~ K'.

the

above.

dia­

same

See

as

[Rl]

point.

to v e n t u r e

out

into

the w i l d

safely!

Suppose

indicated

write

if a n d

K = K'

via Reidemeister

the T h e o r e m .

w i l d kn o t

tame kno t r e p r e s e n t e d

that

K # K'

# K # K'

#

= (KUK')

(K#K') #

•••

3 ( ^ 0 -

Form

below:

% = K U K'

ar = C )

~ 0.

#

# O

# 0

•••.

# •••

Hence

the

CHAPTER IV

58 On

the o t h e r

hand,

* = K #

(K'#K)

#

(K'#K)

#

•••

* = !(#&##•••=:# .*. Hence,

by F a c t .

Re ma r k.

This

an d

ar e

Nn

~ K.

sam e

Mn # Nn = Sn

(remove resulting

and

as Let

the

R

is c o n n e c t e d

Find a too

is

Mn

£

then

denotes

for m a n i f o l d s

together

along

the

idea of g o i n g

off

live w o r k s

of

formula

R s u ch

for

into

in some

place

other

r i ng w i t h u n i t

that

(1-ab)

some

^

1-ba

r e a lm s 1.

is in v e r t i b l e .

thereby showing

i nv e rt i b l e .

Solution:

(1-ab) ^ = -r~~— tt v J 1-ab = 1 + ab + a b a b

+ ababab

=1 + a ( l + b a + b a b a + • • • )b .'.

Exe r c i s e .

if

manifolds,

where

sum

be a n o n c o m m u tat ive

a , b be e l e m e n t s

it

Nn S Sn

that

■

boundaries).

Let

P ro blem: that

#

the proof.

to s h ow

n-dimensional

and

" i l l e g i t i m a t e ” limits

well.

completes

f ro m e a c h a n d p a s t e

Furthermore, where

closed

Mn = Sn

n-balls

This

i de a c a n be u s e d

compact

homeomorphism,

X = K.

Show

(1-ab)

that

this

* = 1 + a(l-ba)

formula

r eally

*b.

works.

+•••

CHAPTER

Example

k .7.

A

Thus

B

A

and

Example

k.8:

available to

tang le and

T hu s

A'

are

T a n g 1e T h e o r y .

[Kl],

is w o r t h w h i l e theory.

"outputs"

C

skein-equivalent.

for p o l y n o m i a l

[ G l ] , [Cl], It

59

IV

A in

[K2]

There

calculation. and

[LM]

form:

many We

the

diagram

more

r e f er

for m o r e

to e x p l a i n he r e

tang 1e is a k not this

ar e

tricks

the

re a d e r

information.

elements with

of

"inputs"

CHAPTER

60

is a s i m p l e connected

tangle.

Assuming

in w i t h a

larg er

IV

that

k not

the

t an g l e

d i a gr a m,

yo u

is a c t u a l l y ca n d e c o m p o s e

i t ske in-wi se .

Here

we

t a ng l es

have

that

above

Clearly, »8=2 ’ * * * cates

T = £1 ©

Q.

and

the

skein

a n d by

the p r e s e n c e

unlinked Now

Thus

circles we

®

for of

of

many

the

we

reasons

tangl e s

= -= ^ '=2 1 * * *

in

define

where

(n

have

called

about

to be

is g e n e r a t e d where

for

the

0^)

the

two

re v ea l ed .

by

subscript

indi­

unknotted,

ta ngle box.

addition

of

t an g le s

in

the

obvious

way:

0 +

“ + “ =

F in a l l y ,

we

1 inks

tang 1e s :

to

define

^ two

=I^> & operations

that

C !l = ® j .

associate

knots

and

CHAPTER

IV

61

This

is d e n o t e d

N(A),

and

called

the n u m e r a t o r

This

is d e n o t e d

D(A),

a nd

called

the d e n o m i n a t o r

We a l s o of

A,

define

F(A),

by

a quotient

o f p o 1y n o m i a l s

of

the

A.

of

A.

f rac t i on

the

F( A) = v NA/ v DA. Thus,

the n u m e r a t o r

tang le

ar e

and

the C o n w a y

denominator

of

THEOREM

( C on way).

4.9

you

polynomials

the

The

Example:

the

fraction

fractions:

the

fraction

the n u m e r a t o r

of a sum

of

the

an d

= V N A V D B + V D A V NB

VD(A+B)

= VDAVD B ’

fraction

0 + ® =

is

the

Explicitly,

VN(A+B)

— + — = xw+yz y w yw

like,

of

of

A.

(f o r m a l ) sum of

N ote:

denominator

formal sum of fr a c t i o n s . ------- --- — -----------

x

—

»

is an

ordered

pai r

[x,y ]. )

fIf v

62

CHAPTER V CD

IV

A

= Vf- = T FJ.) - I S = I

Example:

0

1

1

0

(-)

I 0

A little

Thus,

1- 0

" 0

®

f

Conway's

0-0+1-1 _ 1_

_

= £.

I

+ 0

theorem for

its proof

1*0+0*1 _ o 0*0 “ 0*

“

thought

F (“ i) = §

shows

that

it is sufficient

to check

the skein generators as we have done!

is easy.

But

it is a powerful

tool

for

c a 1c u 1a t io n .

Note: since

T =

^ = -3z,

should be called

= 1.

Call

So

r ( |_n J ) = nz/1, But now

M L nJ

) = 1/nz.

this

[2]"

[~3]

CHAPTER

63

IV

1 _ 3z^ 2z 1

F (A ) = F ( w V F ( r - c ) =

1 -6 z 2z

2

A v

v

Thus

we

conclude

that

In g e n e r a l ,

the

quickly

calculate

inators

of

tan gl e t an g l e s

[n]

i n v e rs i on ,

K = NA

fraction

tang 1 es

by a d d i t i o n

(n € Z ) .

n a m el y ,

2z .

DA

polynomials

rational

obtained

if

1-6 z 2

NA

theorem of

We h a v e

allows

the n u m e r a t o r s

where and

(4.9)

a rational

inver s i on already

and

the

one

In g e n e r a l ,

is a

i nt e g r a l

example

[n]

-i [i]

to

denom­

t a n gl e

fro m

s ee n

us

[i]

of

CHAPTER

64

Note the

that

A

i n p u ts

^

and

is o b t a i n e d o utputs)

upper-left/lower-right

N o w we h a v e

that

FCA"1) =

1/ F (A ) .

For

example

see

we

A

rotating

(up A

to by

isotopy 180

o

that

fixing

around

the

diagonal.

*)

ca n

= D(A)

for m

and

continued

D (A

^ ) = N(A).

f ra c t i o n s .

z^ + 2z

T hus

We

N(A

by

f rom

IV

Hence

CHAPTER

IV

65

-+1

since

this

is

the

numerator

■

Exercise. Let --------------

C

[

of

1]

+

= N n

F ([1]+1/[1]).

i

And

---------------------- •

[ 11 + -------- :— L J 1 • • • -f---- -

with

n

[i]

appearances and

links

of

[1].

Show

are defined

that

~ K n +i

in Example 4.1.

where

the knots

66

Just

CHAPTER

as

fraction

the

snail

knots

builds

spiral

his

IV

shell,

ou t wa r d s ,

so do

these

continued

Rational lens

spaces O ne

e qual yet

to

([ST],

more 1.

(unless

knots

an d [SI])

e x a m p le . K

CHAPTER

IV

t a ng l e s

a re

as

you've

do n e

important

branched

A knot

is a c t u a l l y

67

K

whose

knotted,

Exercise

covering

but

in s t u d y i n g spaces.

polynomial we

can't

3.1 5 !)

K K = N (A +B + C)

F(A)

= 1/z,

F(B)

= 3z/l,

F(C)

= -3z/l

is prove

it

68

CHAPTER

You

should

N ot e

that

Hence

this

F (A +B+C) • •

Remark: foil

check

that

is

the

= F(A)

V N (A + B + C )

The

knot.

Exerc i s e .

l in k i n g

number

of

=

the

two

curves.

+ F(B)

+ F(C)

1/z,

of

is c a l l e d a cab 1e of

=

numerator K

IV

i tself

C

is an u n t w i s ted d o u b 1e of

the

out

of

(a)

Work

Exerci s e .

Take

a k n ot

cable

reverse

the

t h e or y

of

t an g l e s

the

tre

trefoil

the

form

Research

with

orientation.

"

>C

Form

the

CHAPTER

Prove

that

eas y p r o o f proof.]

= later.

lk(K)z. There

[In

IV

this

69

p ro b l e m ,

s h o u 1d be a g o o d

we

shall

see an

skein-theoretic

V DETECTING SLICES AND R I B B O N S — A FIRST PASS

Here

It a

is a r ib b o n

is c a l l e d

a r ib b o n knot

" r i b b o n ” that

with

The

ribbon

ribbon,

pin g the

a form

image

kno t :

is

because

immersed

into

it

forms

the b o u n d a r y

three-dimensional

of

space

singularities:

or

2 : D— » IR

ribbon 3 C S

illustrated

singular

set

disk, 3

whose

above.

consists

is

the

image

a(D

)

onl y

singularities

Thus

each

in a pai r

70

component of

closed

of a ma p are of

of the

intervals

CHAPTER V

in

D

2

*•

one w i t h

end p o i n t s

i nt erior

to

entirely

D

2

on

71

the b o u n d a r y

of

D

2

,

one

.

3 Exercise. clasp

Show

that

every knot

K C S

bounds

singularities.

A clasp

But Th e

not

f irst

we k n o w form?

every

disk

that Read

it

to be in


T ~ 0

is not not

(5.2),

in

the

and

tec hniques

refe r

the

ribbon.

trefoil.

T is

2).

exercise

is not

T

two p a s s -e q u i v a I e n t

exercise

3.

reader

the

polynomial.

5.6.

knots.

of S e c t i o n

the p r o o f

Then

a(T)

=

1

pass-equivalent

T-equivalent we are

done.

to ^

while to 0.

0

78

CHAPTER V

It the

remains

trefoil

to b r i n g

or

to to

in m o r e

PROPOSITION connected oriented

show

the u nknot. geometry.

5.8.

Any

oriented boundary

Proof.

We

Seifert

[S].

Seifert

surf ac e

SEIFERT'S

that a ny

produce

K

is

T-equivalent

In o r d e r

to do

this,

we

to ne e d

In p a r t i c u l a r ,

oriented

knot

3

F C [R

surface

or

K

link

s uch

bounds F

that

a

has

K.

the

surface

Accordingly, (when

ALGORITHM

1.

D raw

the p l a n a r

2.

Draw

the

it

the

by an a l g o r i t h m

due

surface

called

is p r o d u c e d

will

by

this

(illustrated). diagram

corresponding

for y ou r

universe

knot.

k.

be

to H. the

algorithm).

CHAPTER 3.

Split

every

crossing

of

79

V

k

in o r i e n t e d

fashion:

V The

resulting

the S e i f e r t 4

.

5.

set

circles

Attach

a disjoint

to

Seifert

the

Between

of

disjoint for

i s the

of

One

disks

disk per

split-points

add

according surf ace

O r i e n t a b i 1 ity if y o u

curves

is c a l l e d

above

the p l a n e

K.

collection

circles.

closed

to

the

circle.

twisted

crossing

in

bands K.

This

F.

follows

That

is,

start

pass

through a number

from

in a g i v e n of

t he J o r d a n domain

crossings

in

in

curve

theorem.

the p l a n e

the S e i f e r t

and

surface

80

CHAPTER V

o nl y

to r e t u r n

even number surf a c e

diagram

surface by

traversing

to a v o i d

disks

ca n be

crossing

disk

draw.

y o u m us t

This

pass

geometry

t h r o u g h an

passes

over

to

the

may

sketched

Seifert

directly

circ l es :

on

the k n o t

jump at

each

cross­

it!

Circles

be n e s t e d

K The

d o main,

crossings.

Se i fert

The

that

itse lf .

T he

ing

of

to

as

in

k

corresponding

to

Se ifert

this

"fringe"

Circles

is not

so ea s y

to

CHAPTER V Exercise. the

D ra w a g ood p ict ure

figure-eight

kn o t

for

orientable

to recall surfaces

For a s i n g l e b o u n d a r y is a c o m p l e t e surf a c e s . the

list

(See

the S e i f e r t

su rface

the c a n o n i c a l

(abstract

c om p o n e n t ,

E a c h of

representatives):

(F q , , • • • } ,

of h o m e o m o r p h i s m

[MA].)

representa­

types

these

of

Fq

= D

2

,

orientable

representatives

takes

f orm of a d i s k w i t h a t t a c h e d bands. Up

to a m b i e n t

i so t o p y . an e m b e d d i n g

of

F

looks

S

a s t a n d a r d 1v e m b e d d e d di sk wi th twi s t e d . k n o t ted a n d bands. fits

for

(above).

N o w we a l s o n e e d tives

of

81

(That

into a

embeddings F (1) C IR

is,

gi ve n any

time-parameter F {t)

meets

such

that

3

embedding f a m i l y of F C IR^

F C IR

is

the d e s c r i p t i o n a b o v e . )

F(0) For

1 inked

the n

continuously C IR3

1 ike

F

changing an d

e xample:

CHAPTER V

82

Here

we h a v e an e m b e d d i n g as d i s k - w i t h - b a n d s

is a

trefoil

Exerc is e . ambient

Now bands of

Show

that

i sotopic

the S e i f e r t

to

l e t ’s call

t wists per b a n d

(Compare Exercise

F

surface

for

the

trefoil

is

above.

a disk with

twisted,

Since

(o r i e n t a b i 1 i t y ),

k no t t e d ,

t here

lin k ed

is a n e v e n n u m b e r

t hese

can be a r r a n g e d

2.7.)

the b o u n d a r y ,

a pass-move:

boundary

knot.

a standard e m b e d d i n g .

On

wh ose

e a c h curl

affords

the o p p o r t u n i t y

for

83

CHAPTER V

T hi s

means

m o d u 1o

two

Since an d

linking

k no t the

bounds

that

we_ can

reduce

it

is o b v i o u s

of b a n d s a

that

we

rid

by pa s s i n g ,

surface

that

cu r l s

per

ba n d

Hence

we h a v e

LEMMA

5.9.

can get we

of k n o t t i n g

can a s s u m e

is a b o u n d a r y

that

connected

our sum

of

cu l p r i t s :

trefoil

Any

of

by b a n d - p a s s i n g .

following

foils.

the n u m b e r

unkno t shown:

Any

knot

knot

is p as s - e q u i v a l e n t

is p a s s -e q u i v a l e n t

to a sum

of

tre­

to

its m i r r o r

image

all

the band s!

^

(K ? K !).

P r o of :

Th e

To

see

the

c o u p de

PROPOSITION

5.10.

last

gr a c e

part,

p a ss

is:

For an y

knot

K,

K # K!

is a r i b b o n

CHAPTER V

84

Proof :

Connect ribbon

Proof s ho w is

corresponding surface

true

across

the m i r r or ,

and

the

appears.

of T h e o r e m that

points

5.3;

K # K ^ 0

in g e n e r a l

By

the d i s c u s s i o n ,

when

s in c e

K

is a

K j K'

it

suffices

trefoil.

to

In fact,

this

i m p l ie s

K # K ~ K # K ‘ - 0 s in c e

ribbon

COROLLARY

implies

5.11.

Two

( T -e q u i v a l e n t ) if a n d w here

Remark: ca n be A RF(K ) ,

a(K)

We

implies

K,

knots, o nl y

if

£ 0.

K', a(K)

|

are

pass-equivalent

= a(K')

(modulo

2)

= a^(K).

shall

identified of

j: 0

later

see

w i t h w ha t

the k n o t

K.

that

a ( K)

is c a l l e d

a(K)

tak e n m o d u l o the

is e a s i l y

Arf

two

i n v ar i an t ,

calculated

using

CHAPTER V

the

techniques

culated

a(K)

As a result,

Exerc is e . is a 3 - 4

It a

= 1

for

we k n o w

Show torus

is c a l l e d

a

that

III.

Thus

in E x a m p l e

3.8,

we

cal­

K.

that

the

this k n o t

is not

f o l l o w i n g k no t

ribbon.

is not

ribbon

(this

knot).

torus

k no t

because

it

lies

on

the

surface

of

torus.

Remarks

on S l i c e

s l i ce

(in

s l i ce

knots

be

of C h a p t e r

85

the

fact, that

Knots. r i bb o n) are

not

s t e v e d o r e 1s kno t :

We h a v e

s een

for a n y k n ot of

this

form.

i K # K'

that K.

There For

are

ex a mp l e,

is many let

K

86

One

CHAPTER V

can

see a m o v e

K

K

o t h er

shows

Take

d i s k as

b efore

a disk with

you

one

way

two u n l i n k e d

a bit.

knotted

slice

follows:

just

sadd1 e

bounds

T hi s knots:

the

just

s tevedore

Hence

for

it be,

Go

singularities

to c r e a t e

c ir cles.

th r o u g h a s ad d le

then

after

sa d d l e

u n 1 in k e d

'tis a slice

volleys

Wrap to

knot

in

the

of

form

s lice

them a r o u n d

form a knot. for

1 i nk

sure.

If

e ach

87

CHAPTER V

E x a m p 1e :

E x e r c i se 5 . 1 2 . it has

Prove

a movie

with

that

a k no t

is

saddle

p oints

an d

ribbon

if a n d

mini m a,

but

o nl y

if

no

m a x im a .

Exer c i se 5 . 1 3 . (not

Prove

a connected

THE KNOT Two cordan t

CONCORDANCE knots

K,

(K ~ K') F :S 1

ding

unkno t . ponent

is

3

K'

C S

if

there x

I

represents an y

Just

the m o v i e

left!

Knot

is p r i m e

kn ot s ).

GR OUP

ex a mp l e, run

the S t e v e d o r e ' s

two n o n t r i v i a l

such that

an d F |S ^ x 1 For

of

I — ♦S 3

x

uni t i n t e r v a l ) K,

sum

that

s lice

are

said

exists

to be

a differentiable

( I ={ t € IR

1

F|S

x

(smoothly)

| 0




= F ( ^ , )

- F ( ^ » )

Ff

) (drop m u l t i p l e

- F(

ed ges)

= F(. N ) - F(*v^) - F( ^ ) 4*12

- 12 - 12

24. Certainly, ula bear

a

these

f ami 1v re lat i o n s h i p

kno t no 1v n o m i a 1 s v i a dix

for a d i r e c t Wh a t

sibility planar

c a 1c u l a t ions

does of

map,

the

F(G)

w ith

the C o n w a y

relationship chromatic =0?

If

then we w o u l d

some

s mallest F (G - a) a nd the

t

r

Thus

a

of

same x n f-

the

in

G-a

color and

so

means

y

ar e

^

four-color

calculations (See

the J o n e s tell

for m of

the a p p e n ­

polynomial.)

us

about

the p o s ­

a n o n - f o u r - c o 1o r a b 1e

- F ( G /a )

Let's that

that

(formerly in e v e r y

was

chromatic

ha v e

G.

noncolorable

= F(G/a)

y

that

ed g e

our

formula G

the

iden t i t v .

with

0 = F ( G -a ) for

with

suppose F ( G -a )

the

the ends

coloring

f o r c e d , an d

G

? 0 ? F( G/ a ).

" exp osed" of

of

is a

a)

G-a

that

•

T =

is:

Then

vertices

always

(G-a)

d_o tyr an t s exi s t ?

x

re ceive

We will

_

CU

problem

two

that

say

is a

149

CHAPTER VI It After

is,

all,

ordinarily switch

in fact, if

T

good

is a

of

they m u s t

to o b t a i n a n e w Let's

try

tr y i n g

tyrant,

communicator

the c o l o r

q u en c es ,

very

x

and

to

then

start

eventually

and

to m a k e

a nontyrant,

such

Thus,

a

switching

(of

The

two

3 other

vertices.

colors for

y

x

as

if y o u

following y

If we

d o w n the c o n s e ­

as well,

#R___ #B #

s witch forced

eventually c an be a

mus

if

T

is

#R y

x's

red (R) to

to ch ange,

f lips

y

tyrant

in a v e r y

to

them

we 11:

about a

t be c o n n e c t e d

Otherwise

affecting

connected

among

y.

extra­

tyrant:

thi n g we n o t i c e

is that

directly

and

an

tyrant.

to

B

B

R,

as well. limited

c ol o rs ) .

fir s t

without

x

switch

that a r e

graph

colors)

c o l or

but

B' s

the a b o v e

domain

is a l s o

a

co l o r i n g .

switch any

then

to c r e a t e

T

between

#R__ #B__#R____ #B__#R__ #B x # # Here's

try

to (for

y. x the

tyrant (for

d i r e c t l y toat

x

would

be

And

these

3

must same

a lways

free

reason).

three All

least

to c h a n g e

(or m o re )

have

four

vertices

different

this

is

true

CHAPTER VI

150

We

could

Suppose

x

try

to do

this as

is c o n n e c t e d

s i m p l y as p o s s i b l e .

to e x a c t l y

3 other

vertices:

1

If we ored

The

try

differently

second

f rom

x

We've

firs t

r es u l t bring

of

sin c e

ar e all

them we

there

co l o r

into

d o e s n ’t look

further

the map.

is d i s a s t r o u s

2 back

(1,2,3)

connecting

is out

possibility 2 fr o m

nether-reaches

the

these

mu s t

col­

find:

ex i s t

a path

y.

dropped

change

that

by d i r e c t l y

possibility

to

Th e

we

to e n s u r e

at for

x

direct For

communication

with

the

exa m pl e :

without tyrants!

communication:

too p r o m i s i n g .

disturbing

1 or 3.

Of

we

course,

can

The try

to

151

CHAPTER VI

4

But

this

splits

the c o m m u n i c a t i o n b e t w e e n

1 and

3,

allowing

B R.

with

x

free

Perhaps ing a

begin

Proving

a dangerous is du e

technique

colors. to see that

game.

the d i f f i c u l t y

t y r an t s The

be s t

don't

exist

guarded

to H a k e n an d App el 1, u s i n g

in c o n s t r u c t ­ is,

proof

much

at

of the

computer

[AH].

Exerc is e . below?

you

tyrant.

course, moment

to c h a n g e

Ho w m any

Sa y h e l l o

I sometimes

dream

substructure.

Ah

to

four

colorings

the n o n p l a n a r

of p r o v i n g s o . ..

ar e

there

for

the g r a p h

tyrant:

that a n y

tyrant

has

V

as a

152

CHAPTER VI

§17.

THE MOBIUS Th e

Mobius

BAND band

is

usually

represented

by

a

drawing

s uc h a s :

Si nce

the b o u n d a r y

research

problem

in

e m b e d d ings

of

For

a good

history

of

Here

follows

an

(her

the

ofthe b a n d

band

is u n k n o t t e d

it m a k e s

three-dimensional graphics with

s tandard1 v

this p r o b l e m , original

nice

to

f ind

u n k n o t ted b o u n d a r y .

see

solution

a

[S2]. by

Carmen

d r a w i n g s ):

U s u a 1 M o b iu s

Twi s t O v e r

Safont

CHAPTER VI

Straighten

1-2-3.

and b e g i n

153

to smooth.

somewhat

more

rounded

154

CHAPTER VI

piecewise Here

the

simplices

462,

417}

f orm

the b o u n d a r y .

fo r m

{156,

the M o b i u s

652,

257,

Band.

751,

Edges

linear

136, 14,

237, 24,

467,

23,

31

155

CHAPTER VI

Another computer tion of

possibility

to p r o d u c e the

= {(z^.Z g )

L.

Siebenmann,

representation

drawings

following

S3

for

of

the

embedding

| | z 1 |2 + |z2 |2 = D a n As i mo v ,

of

1}:

is

the u se

stereographic the M o b i u s

(This

has

projec­

band

been

of a

in

done

by

the a u t h o r , . . . ) 7r

[ 0 , 2 tr] x

TT

2’ 2

M( 0 ,4>) = ( cos ( )e

i0 / 2

i0

L. S i e b e n m a n n & A s s o c i a t e s , 1984

§18.

THE GE NER AL IZE D POLYNOMIAL An

during t hese

extraordinary the S u m m e r

n otes

prepared). invariant

are

of

1984

based,

Vaughan of

breakthrough

and

Jones

oriented

(just

after

during [J01]

knots

a nd

occurred

the

in kn o t

the c o u r s e time

on w h i c h

they w e r e

discovered

a new

links

satisfied

that

t heory

being

polynomial a set

CHAPTER VI

156

of a x i o m s

as

follows:

AXIOMS 1.

To

each

s uc h

Vv = V „ , K K

whenever

that

=

" Co nway

t V

= (>Tt -

polynomial"!

stranger

into

of a n e w

than

via a cons tr uct ion

of

new polynomial many

images.

previously It Jones' end Ju ly

and

K'

are

whenever

K,

K

For

t heo rists

this

and

and has

it has

knot

He

in

the

sky.

obtained

representations algebras

a power

of

the

that no

from

simple

classical

u sed

mainly

Furthermore

ability

t re foil)

Jones'

polynomial

physics!

incredible the

his

previously

statistical the

An d

was

to d i s t i n ­ their

mirror

invariant

had

assumed.

is b e y o n d

the

representation

there. of

new

kn o t s ( i n c l u d i n g

Thus

K

1/nTT)VL

star

that.

operator

in q u a n t u m m e c h a n i c s

guish

e Z[t,t

r e l a t e d as

the a p p e a r a n c e

this

is

1.

are

group

there

iso t op i c.

-

L

braid

K

link

polynomial

t

w o r k was

or

a Laurent

3.

like

POLYNOMIAL

associated

V

Another

JONES'

o r i e n t e d knot

ambient 2.

FOR

1984

Many

capability theory.

people

heard

(the p r e s e n t

of

these

Ho w e v e r , Jones'

a uthor

alas

notes

the

init i al not

to go

story

does

lectures

among

them)

into not in and

157

CHAPTER VI

a mong

these

a number

embracing both

saw c l e a r l y

the C o n w a y

g e n e r a 1 i zed p o 1v n o m i a l . dently Pe te r

by:

Ken Millet

Freyd

and

[ H O M F L Y] ) . tive of

Of

technique

David these

dently

Yette r , peop l e,

theory. by

was

Adrian Ocneanu all

Ocneanu

Joze f

forth

Lickorish,

did

or b r a i d

The

polynomials.

brought

Raymond

on d i a g r a m s

discovered

J o ne s

G^ ,

an d

F r e y d - Y e t t e r ) except

representation

an d

to a g e n e r a l i z a t i o n

their

H.

Ji m Hoste, (see

Przytycki

was

also

an d

by

(in

who g e n e r a l i z e d

polynomial

the

indepen­

work

diagrams

Thus

induc­

the

cas e

the b r a i d indepen­

Pawel

Traczyk

[PR]. The

generalization

Conway Jones

(For and

yet

Question: the

as

(A)

vk

— ■ ->

(B)

t_ 1 V R - t V^- =

discussion

to

about

— .. >

another gen eralized

further

Appendix

comes

these

For

of

f ollows:

“ VK = ZVL

polynomial

the J on e s

[lT - ^ = ] V L •

du e to

polynomial,

see

the

notes.)

what

"arbitrary"

coefficients

re lat ion

a G „ + bG pr = K

K

the au t ho r ,

K

K

c Gt L

L

a,b,c

does

158

give

CHAPTER VI

an

invariant?

normalization

the

And

situation

AXIOMS 1.

To e a c h iated

i nc r e d i b l y ,

oriented

knot

a Laurent

K

K

and

or

by

After

the

POLYNOMIAL. link

polynomial

there

in

two

€ Z[a,a ^ z , z

are

does.

is a s s o c ­

variables

*]

a m b i e n t is o to p i c ,

then

= G K-

gk

2.

=

3.

expressed

FO R G E N E R A L I Z E D

G^( a,z) If

ca n be

it a l w a y s

aG

1.

^ - a

*G-y*

= zG—r . •------------------------------

Thus:

(i)

a= 1

(ii)

■ > G^ =

z = I— - - >fa

, = >

the C o n w a y

polynomial.

G^ = V ^ (a ) ,

the

J on e s

polynomial. Other

authors

z = -m

gives

What

different

ar e

many

in

o f kno 1 1 e d . t w i s t e d

bands

The new va riable

axioms

are:

20)

to a s k a b o u t

the ne x t

interpretation.

(by S e c t i o n

of L i c k o r i s h

questions

the g e n e r a l i zed p o 1 y n o m ia 1

shall

Thus

a = H

*,

a n d Mi l le t .

on h e r e ?

I take a ste p

geometric

lett e ri n g.

the p o l y n o m i a l

is g o i n g

There iants.

us e

What

two

these

sections

I s h ow

is

new

toward a

that a ver s i on o f

i s an a m b i en t iso topy in

th r ee - di me n s i o n a 1

measures construct

twisting a

invar­

in v a r ian t

space.

of bands.

B-polynomial

Thus,

whose

we

CHAPTER VI

AXIOMS 1.

Let are

K,K'

FOR THE

159

B-POLYNOMIAL

be

oriented

links

(knotted)

oriented

t w i s t e d bands.

times

called

fr amed

ambient

isotopy

space.

Then

B^fa.z)

€ Z[a,a

of

whose

I i n k s .) bands

there

in

^

Let

^1

(Some­ denote

three-dimensional

is a L a u r e n t

*,z,z

components

such

polynomial

that

K

B v = B v/ K.

K.

K ^ K '.

whenever

aB = a _1B

Note form ing

that

the

exchange

(no n e w v a r i a b l e s ) . in

The

ide ntity

is

in s t a n d a r d

n e w var i a b 1e m e a s u r e s

twist-

the b a n d s .

By u s i n g lar

here

isotopy"

topological (see S e c t i o n

the c o n n e c t i o n

between

the

s cr i pt 18)

and

the

the n ext

concept

two

B-polynomial

of

sections

and

"regu­ draw

the g e n e r a l ­

ized p o l y n o m i a l . For sample

the

rest

of

B-polynomial

this

s e c t i o n we go

cal culations:

t h r o u g h a se r i e s

of

160

2.

CHAPTER VI

By

the a x i o m

B

3:

B

= zB

$ -i

@r

.

= B

Le t

S = (a-a

^ )/'■

BL = 6 BU =

= Bj- + z B y = 6 + a

(a - a

4.

* )z

W>

* + a

*z .

mm a B

bk

=

bk

bk

= “

“

bk

(1 -a It

is

ea s y

to

bk

_ zBL

)Bj^ = z Bl . see by

( 6 = (a-a 1 ) / z ) t

induction

where

that

L = A LIB

B^

= S B^ B g

is a spli t

link.

161

CHAPTER VI

R 1/

H en c e

K

—

Za

_a-a

i -1

If

-l] a ~ a

J[

z

R -tS »UR ti

J

A B

Bk = “ b a b b and

bk = “ ’ 1babb-

A similar gives

argument

the p r o d u c t

s hows of

that

a s traight

5.

B

B

n

n

= B„

= B

. i\ n

sum’

the p o l y n o m i a l s :

A*8

Let

"connected

Then

0 + zB n-2 n-1

B„ =6, u

B. = a, i

b a #b

-

baab

162

CHAPTER VI

1 (a-a 1 )

Bq

=6

= z

Bj

=a

= a

B^

= za + 6

= za + z *(a-a

2

2

B ^ = z a + z 6 + a B^

6.

= z^ a + z ^6 + 2 za + 6 =

It

follows

K

by

it

is u s e f u l

is not

This

link g o e s

equ al

to

for

Thus

B^(-a

the

by w a y

of

if

*) + z * ( a - a

i K*

is o b t a i n e d

then

B

i nvariant

we

see at

*)

from

f( a fz) K* under

a — *• -a

in

f o rm

the

on c e

*

of

that B ^ ( a , z )

*,z).

f l a vo r

t w i s t e d bands.

con nection with

)

the p o l y n o m i a l s

z,a,6 .

of

is

6

leave

should give

polynomial the

to

that

crossings,

Since

1

(2a - a

z^a + z ( 3 a - a

induction

*,z).

functions

e ate

by

r e v e r s i n g all

= B^(-a

-

= z a +

*)

of

calculating

Th e n e x t

the g e n e r a l i z e d

topological

script.

two

the

sections

polynomial. If we

denote

Bdelin­ The

163

CHAPTER VI

and

Consequently,

the

type

§19.

I

f

--

we a r e

ar e

led

to c o n s i d e r

> o ^ v > o

the

crossing

Define

the wr i t h e .

of all

of

its

isotop i c.

diagram moves

without

AND R EG U L A R

ISOTOPY

signs:

w(K) ,

crossing

ambient

replacemen t .

T HE G E N E R A L I Z E D P O L Y N O M I A L Recall

not

of a d i a g r a m

K

to be

the

sum

signs. w(K)

= I e(p) P

where

p Thus

writhe

runs the

of

+3

In g e n e r a l , switching

over

tref oi l and

a knot

all

all

the

-3,

and

crossings T

a nd

in

the d i a g r a m

its m i r r o r

image

K. I T*

have

respectively.

its m i r r o r

crossings)

h av e

image writhe

(obtained of

by

opposite

sign.

164

CHAPTER VI

If then

the w r i t h e

its

calculation

tinguishing re gular

mirror

is otopy

regularly a

were

in variant

of a m b i e n t

w o u l d be an e x c e l l e n t

images.

(denoted

isotopic

sequence

an

Writhe ~) .

if one

of R e i d e m e i s t e r

i_s^ an

Tw o

can be moves

method

obtained type

for

invariant

diagrams

of

isotopy

are

from

of

sai d the

II or

dis­

to be

other

type

III.

II .

Ill .

Generators

Thus

the w r i t h e

t h e or y

of

regular

writhe

is not

of R e g u l a r

is an

isotopy.

regularly

Isotopy

invariant An d any

isotopic

Unfortunately

(or

simply

stated

problems)

the a m b i e n t

mirror

i ma g es

is not

simple.

of

moves

mirror In

II,

III

fortunately

to

and

so

also

(!)

in

" f l a t ” kn o t

diagram with nonzero its m i r r o r

if y o u

love

isotopy

Perhaps

I will

the

turn a

image. deep

and

problem

some

for

combination

trefoil

into

its

i m age? fact,

this

is not

so!

The

trefoil

is not

ambient

by

C HAPTER VI

isotopic

to

later

the b o o k by

an

in

its m i r r o r

i nva riant

that

image.

the

polynomial

is an

related

the g e n e r a l i z e d

to

(We

signature

I call

invariant

165

shall

prove

methods.)

Here

R—p o ly n o m i a l ,

of

regular

this we

R^.

isotopy,

polynomial

Gv K

by

again

construct The

and the

R-

it

is

formula

GK _~ a-w( K)R K where

w(K)

is

the writhe.

AXIOMS 1.

To

each

o r i e n t e d knot

R„

K.

or

link

ated a Laurent

polynomial

The

is d e n o t e d

polynomial

€ Z[a,a

R^(a,z) It

2.

F OR

is a n

R ^

in

there two

\z,z

of

K ~ K7

— ■> Rk = Rk , .

= R ^

variables

a,z.

*].

inuariant

regular

is a s s o c i ­

i s o t o p y .*

= 1,

o

R — >0'”+ = aR ,

3.

Remark:

R

= a ~ 1R.

R

- R

As

with

abbreviations cated

patterns Unfolding

gram

s uch as

= zR

the C o n w a y

refe r ar e

to

polynomial,

larger

diagrams

these

diagrammatic

in w h i c h

the

indi­

e m be d de d .

Axiom

2,

we

see

that

for a c u r l y

unknot

dia­

166

CHAPTER VI

the R

U

R-polynomial

the w r i t h e

w(U)

in

the

f orm

- a W (U > “

The for

returns

th i rd a x i o m

the C o n w a y

no mi a l.

Not

looks

just

polynomial.

But

at all!

View

R t, = aR ^

K = a

We

see

that

polynomial.

*),

components. of

E xerc is e . split

B

unlink

R, it

of

that

if

= z

an u n l i n k is u s e f u l of

reader

of S e c t i o n

Show

R,

rk

with

the

following

the C o n w a y

poly­

calculation:

= a

the v a l u e

[As

is not

identity

"I

R6

Rk

In fact,

6 = z * (a-a

version

=

r l

R

the e x c h a n g e

= a

“ 1D

•n

2

the

like

R

c an

(a-a

can

).

receive

a nonzero

to r e c o r d on a

see,

spl i t R

is a

unlink

of

two

script-

18.]

= 6

n-components

6 0

then

••• R^

0

= 6n n

denotes ^.

a

CHAPTER VI Exerc1 s e .

Show

that

R

,(a,z)

= R^f-a K

K the m i r r o r - i m a g e s u r i n g all

The

diagram

that

167

-1

,z)

is o b t a i n e d

1

where

f rom

K ‘

K

is

by m e a ­

the c r o s s i n g s .

generalized

p o 1y n o m i a 1

G^(a,z)

is d e f i n e d

by

the

e q u a t i on

Since

R

that

Gj^

is

= a R ^ an d

R—

= a

invariant

under

moves

the g e n e r a l i z e d topy.

polynomial

[We be g p a r d o n

o rder!

G

ca m e

consistent.

Exerc is e .

first,

for a

Show

the e x c h a n g e

logical

that

that

The foil.

G

,(a,z) K *

for

T

Hence

of a m b i e n t and

here

and

iso­

logical

that

define

R R

is by

path.]

polynomial

satisfies

c om e

= zG — w

- a

= G„(-a K

calculation

calculation

G

once

identity

time has

The

with

prove

at

III.

historical

do not

start

II a n d

invariant

the g e n e r a l i z e d

aGy^ Show

I,

reversing

a n d we

[K 8 ] , or

Se e

R = aW ^ ^ G

for

is an

^R.— ► j**' f ° l l ° ws

*,z).

to c o m p u t e

R

and

exactly parallels

G the

for

familiar

:

T

the

L

tre­

CHAPTER VI

168

Thus T = .* .

T © L ,

L = L

T = T 0

(L © U ) ,

.*.

= RjjT +

Since

R^

clude

that

= a,

0 U,

Z ( R ^ + Z Ry ) •

Rj-= 6(=

z

* (a-a

and

Gt „

. . Note ----- » -a a i 1 ent

to

(2a - a

-2 G^, = (2a -a

*.

G^

k no t

con­

A z 2a , - a -1 +

= a _3RT

-4. 2-2 ) + z a is not i n v a r i a n t tha t the

u n d e r the s u b s t i t u t i o n trefoil

i s no t equ iv a -

image.

is a n o t h e r

figure-eight

we

+ z^a

Thi s p r o v e s

its m i r r o r

Here

*)

= a _ W ^T ^RT

that

R^ = a,

= a + z6 + z a = a + a

R^, =

an d

2

R,p = a + z(6 + za)

.' .

*))

K:

sample

ca lculation.

This

time

for

the

169

CHAPTER VI

H er e

an d

K = K €

L - L 0 U.

Hence

K

(L 9 U)

with rk

= “ _S

r

=

l

a 6

R — = aa

-1

1

.

Thu s rk

=

a

Note the

that

w(K)

-2

this

kn o t would What

shall

A

,

+ a(a-a

ha p pe n . not

prove

) - z

2

R„ = K

to

that

is

this

Since

its m i r r o r

R„(a,z) K

so o b v i o u s to

for

K

R v ! =R v . K K

isotopic

iso topi c

diagram the

that

its m i r r o r

the

Let

writhe.

that

K

we

knew

^,z).

f igu r e-e igh t

im a g e .

In

K

be a knot K

diagram

is a m b i e n t

with

fact,

isotopic

ze r o

to

f the d i a g r a m s

figure-

image,

= R„(-a K

of

following

MIRROR THEOREM. Suppose

-L

Al s o

Note be

the

2 Rt

Hence

is a m b i e n t

ma y

z

Rl

knot.

kno t i s r e g u l a r 1v we

z

= 0.

figure-eight

e i g ht

+

rk

and

K‘

are

regularly

iso to p ic .

i K'.

Then

CHAPTER VI

170

In or d e r

to p r o v e

another

invariant

degree

d(K).

of

the u ni t

the kn o t

of

The

this

regular

Whitney

ta n ge n t

diagram.

r e s ul t

vector

we n e e d

isotopy.

degree to

to

This

measures

is the

the u n d e r l y i n g

Combinatorially

it

introduce

yet

the Whi tney total

plane

is d e f i n e d

turn

curve

of

as

foilows: 1 . d( 4 ) = d( X

splicing

disjoint

= d (X ) + d (Y )

all

the

circuits

of

the

sum of

±1

For

e x am p l e ,

of

of

circles

in

curves

is a d i s j o i n t

in

the plane.

)•

the d i ag r am ,

(the se

the d i a g r a m ) .

in

X U Y

) = d( X

crossings

collection

if

ar e

T he

for

ea c h

Seifert

the

case

of

the

we

obtain

called

Whitney

the

degree

cir c ui t . trefoil:

= >

d(T)

= >

d(K)

-

2.

T and

for

K

the

figure

eight:

W

=-1.

is

a

171

CHAPTER VI

Since crossings e rt y

of

Whitney in

the W h i t n e y

the u n d e r l y i n g and G r a u s t e i n

the

same

We will s t a te

doe s

or u n d e r - c r o s s i n g s

the p lane ar e

have

degree

that

generated

by

[Wl]

depend

it a c t u a l l y curve

proved

upon

measures

i m me r si o n. that

r e g u l a r 1v h o m o topic

two

over­ a prop­

In

fact,

immersed

i f an d

only

curves

i f they

Whi tney d e g r e e . not

it

pl ane

not

explain

regular

is c o m b i n a t o r i a l l y the p r o j e c t i o n s

of

h o m o topy h e r e

except

equivalent

the

the m o v e s

to

II a n d

to

relation

III.

T ha t

i s , by

and

It

is a n i c e

exercise

an

invariant

of

underlying Trick of

(this

this

this

to p r o v e relation

that (see

the W h i t n e y - G r a u s t e i n the

reader

chapter!):

has

the W h i t n e y [Kl]).

Theorem

already

Th e is

degree basic

is

move

the W h i t n e y

encountered

in S e c t i o n

3

CHAPTER VI

172

RH

RK

RK

RK

In o rd e r

to g e n e r a l i z e

the

crossings

Thus

we h a v e

and

the

the W h i t n e y

create

a regular

fundamental

tri c k we

shall

include

isotopy:

cur 1- c a n c e 1 1 ing

regular

iso-

topy :

On

the

o th e r

isotopic regular one

(Pr ov e

isotopy

can prove

invariance

the

Theorem

PROPOSITION.

ly

is no t r e g u l a r l y

to

Graustein

ent

hand

isotopic iso topic

Let to

following (See

K

the w r i t h e . )

generalization

by u s i n g

the

As a result, of

the W h i t n e y -

[ W l ] , [ T R ] .):

and

the un k no t .

if a n d

of

this

o nl y

if

K7

be knot

Then

K

they hav e

diagrams and

the

K7 same

each a m b i ­ are

regular

writhe

and

CHAPTER VI

the

same

Whitney

173

K ~ K 7 w (K ) = w (K 7)

degree:

and

d (K ) = d (K 7 ) .

We

omit

obtained

by a r e g u l a r

of a s t r i n g

and

the p r o o f

of

curls

the a p p r o p r i a t e

when

it

it

is

time

M IR R O R THEOREM. w ri the . K

Then is

Proo f: suf fices ing the

that

isotopy

remark

that

to a n o r m a l

the

for m

result

is

consisting

such as

cancellations

Since

to p r o v e

K

Let K

using

the W h i t n e y

t rick

isotopic

regular that

means

(appropriately

to

that

» K*

and

curly).

Prove

this

» K'

to

is an a m b i e n t given

K

that

ze r o

if a n d

o nly

isotopy,

it

» K ~ K*.

Assum-

is r e g u l a r l y C

where T hu s

C

we h a v e

K ~ K ' # C.

(E x e r c i s e :

with

t K ‘.

» K ~ K ‘

i K ~ K'

diagram

isotopic

isotopy

to p r o v e

sum of

the

be a knot

Is a m b i e n t

regularly

connected

diagram

but

is a p p r o p r i a t e .

Now

if

here,

last a s s e r t i o n . )

isotopic

to

is an u n k n o t

CHAPTER VI

174

Since additive

the W h i t n e y

under

d e g r e e an d

connected

the w r i t h e

sums ( e x e r c i s e )

we

are

see

w(C)

= 0

(because

w(K)

= w ( K ')

d(C)

= 0

(because

d(K)

= d ( K !)).

each

that

= 0)

a nd

By

the p r o p o s i t i o n

K ~ K ‘.

T hi s

Th e

ally

knots

a problem

for

G

li st e d G,

and

in our We between

Theorem f ro m in

the

(or at

R).

The

shows

that

that

proved

the p r o b l e m

reader of

isotopy

the is

this

[ K 8 ] for a p r oo f

for

C ~ 0

and

hence

■

(oriented)

regular

the b e g i n n i n g

to

follows

the proof.

their

We h a v e not

Remark:

then

completes

Mirror

guishing

it

mirror

of

i ma g es

distin­ is a c t u ­

c a te g o r y .

consistency

of

referred

the p a p e r s

section

to for

the a x i o m s

discussions

the

L-polynomial

of a

regular

discussed

appendix. conclude the

with a picture

figure-eight

an d

its m i r r o r

isotopy

image.

of

175 chapter

VI

CHAPTER VI

176

R e m a r k on W e i I - D e f i n i t i o n :

Producing a s tandard unkno t

R e c a l 1 that

in a s t an d a r d u n k n o t . s p 1 i c in g a c r o ss ing

n e a r e s t to b a s e - p o i n t

r e s u 1 1 s in a split

s pl i ce at

uni i n k .

L,

2

p

K This

fact

L

is

the ke y

well-definedness polynomial, To

see

obtained T hu s are

or

the

the

from

and

to a n y

invariance

by

the d i f f e r e n c e that

n

suppose

switching

K = S S .•••S.K. n n-1 1

Assume

of

argument

either

proving

the

the g e n e r a l i z e d

R-polynomial.

issue, K

inductive

1

Assume

between

is a d j a c e n t

that

K

is an u n k n o t

crossings that

labelled

l , * # # ,n.

the c r o s s i n g s

switched

K

an d a s t a n d a r d u n k n o t

to

the b a s e - p o i n t

as

K.

s ho w n

be 1o w : n

|n

— •

>------------— •------------------ »

P

P K

I |

K

K = S S • • -S.K n n-1 1

177

CHAPTER VI n Then

(As

R„ = K

+ z y e (K)RE.S. • • • S 1K. L nv ' 1 l-l 1 i= l

K

in C h a p t e r

case.

3 of

these notes,

It c a n be h a n d l e d

lem ma a b o u t

there

directly

invariance under

is a n o t h e r

or by

basic

first p r o v i n g

cyclic permutations

of

a

switch­

ing e 1e m e n t s .) If we

s li d e

the b a s e p o i n t

n

•

through

the c r o s s i n g we get n

»

•

9

»

1.)

of h a n d l e s

is o b t a i n e d

from

first h o m o l o g y of b o u n d a r y In oth e r in F

components.

f

,

= 1

181

+ 1

the

g roup

components

words,

standard

by a d d i n g

of

the g e n u s form

disks

for to all

182

For

CHAPTER VII

one b o u n d a r y

DEFINITION the

genus

7.1.

c om p o n e n t ,

Let

K,

of

K

2g(F)

= p(F).

be an o r i e n t e d knot

g(K),

Is

the m i n i m a l

or

u a l ue

link.

Then

g(F)

of 3

among

all

sp an

K.

connected,

S i m i l a r l y , the val ue

faces . where

p(F)

of By

our

h(K)

Example: of g e n u s

Is

r an k

among

K

we ha ve

that g(K)

it w o u l d be u n k n o t t e d ) .

is 1 .

K,

of

the n u m b e r

have

surfaces

c o n n e cte d,

formula,

Any knot 1 m us t

oriented

p(K),

F C S

is

oriented 2g(K)

of b o u n d a r y

that

the m i n i m u m spanning

= p(K)

- JJ-(K) +

components

is k n o t t e d an d b o u n d s = 1

T h us

(since the g e n u s

if of

sur­

K.

of

a su r f a c e

it b o u n d e d the

1

a disk

trefoil

kno t

183

CHAPTER VII

Let underlying

U

K

be a k n o t universe

is a p l a n a r

(or

(the p l a n a r

graph with

ces.

By E u l e r ' s

edg es

ar e

Formu la ,

incident

link)

di ag ra m and

let

U

be

its

graph).

R r e g i o n s . E e d g e s . a n d V ver t iwe h a v e

to ea c h vertex,

V-E+R

= 2.

we a l s o

have

Since

4

4V = 2E

184

CHAPTER VII

or

2V = E.

regions

S

U)

denote

(refer

PROPOSITION

7.2.

S

circles F

of

Proof:

F

L et

e^

are

U

g i v e n by

up

There

ar e

two m o r e

K.

g(F)

= i

We k n o w

know

V-E+S

2-R+S

= l-p(F).

that

= V-2V+S H

Pq

= 1,

= -V+S

spanning ha ve

Then

type,

surface

p components,

the r a n k a n d

obtained

w here

f rom

the

to e a c h S e i f e r t

the n u m b e r

e^ = V,

= P 0~"P 1 + P2

K

(R-S-jx) .

one 2 -c e ll

denote

for

formulas

= R -S - l

Then

K

Let

re gions. the

circuits

P r o p o s i t i o n 5.8).

the S e i f e r t

to h o m o t o p y

c o mplex.

of S e i f e r t

p (F)

by a d d i n g

e 0 _ e l+ e 2 = 1,2).

be

R

and

(k = 0 , 1 , 2 )

resulting

5,

link d i a g r a m

is,

1-complex

F

Let

or

Seifert

the n u m b e r

to C h a p t e r

for a knot

g enu s

R = V+2.

than v e r t i c e s .

Let (or

Therefore

e^

of

= E,

k-cells e^ = S.

Pfc = r a nk H k ( F )

p \ = P ( F )» = 2-R+S.

P2 =

Therefore

circle. in

the

We h a v e (k = °* we

185

CHAPTER VII

Seifert

surface

In the

circuit

type

II circu it , the b o u n d a r y solution the g i v e n

1 ows :

is

of

pre s en t .

the p l a n e

supposed

in the usua l

into

or

two

for

link,

(A type

even

II

regions,

this

to a d d a d i s k

drawing,

each

II c ircuit.

circuit"

This

to

the

the n e i g h b o r h o o d

d i s k has an u n c l e a r

to d r a w a "t r a c e r

type

for a k n o t

to

circuits.)

that we are but

indicated a method

surface

II c i r c u i t s

divides

containing Seifert We k n o w

I have

the S e i f e r t

there a r e

Seifert

K is o b t a i n e d by a d d i n g a d i s k the c u r v e a.

fi g u r e above,

understanding when

for

tracer

s t ru c tu r e. a

type of

My

corresponding

is d r a w n as

fol-

to

186

CHAPTER VII

Tracer Thus,

it

follows

necessary. new

become these

bounded regions

that

a

between

neighborhood

the by

the

type

T hu s

the

tracer

of

and

II c i r c u i t

we h a v e

proved:

the k no t

the old

to

is o r i e n t e d it

crossing

traces.

it w h e n e v e r d i a g r a m as a

the dia g ra m ,

I circu i ts .

the di s k a d d e d

a

on

over-passes

tracer

type

II circuit,

d r a w out a p i c t u r e

The n e w c o m p o n e n t to

type

If we d r a w

component

regions

the

Circuit

The

of

type

II

disks

the

circuit filling

the b o u n d a r y

the o r i g i n a l

in

then

of a

type

the o p p o s i t e

in

I.

direction

187

CHAPTER VII

PROPOSITION gram.

7.3.

Let

be

the d i a g r a m ing

tracer

Let

these

from

circuits trac er

only

method

of

representing

Here

is a

Does

every knot

and

fundamental

surf a c e ?

links as we

shall

see shortly.

bound­

i ^j .

for

surface

calculations

later

the S e i f e r t

some

classes

It

is

will on.

s urfaces:

its m i n i m a 1 g e n u s for

K.

on a of k n o t s

fa ls e in gene r al ,

the g e n e r a l i z e d p o l y n o m i a l .

See

the

[M] by H u g h Mo rton.

While ogy b a s i s F,,,. K

of

is true

in

U U * • *U

is a d i s k w i t h

p r o b l e m ab o u t

Th i s

overpass­

circuit

F^,

to

the S e i f e r t

certain

( 1i n k ! a c h i e v e

by an a p p l i c a t i o n paper

for

be

circuits and

D^flD^.=0

And

be p a r t i c u l a r l y u s e f u l

Seifert

I Seifert

(i = 1 , • • • ,k) a^.

ar y

K'

Let

a 1 ,a 0 ,•• • ,a, .T h e n 1 2 k

labelled

is a m b i e n t i s o t o p i c

link d i a ­

disjoint

II S e i f e r t

type

be

type

where

This

by a d d i n g

or

K.

s u r f a c e for

K

for ea c h

circuits

has

(ii)

be an o r i e n t e d knot

the S e i f e r t

obtained

(i)K'

K

Let

Now

we're

at

it,

let's

for Fw K

l oo k i n g at is a p lanar

o b t a i n e d as a c h e c k e r b o a r d It o nl y

leaves

Consequently,

remark

the p l a n e at

on h o w the

sur f a c e . pattern the

trac e r

That

from

to see a h o m o l ­

is,

it

is

the d i a g r a m

twists

is S eneratec* by

surface

c yc l e s

K'.

188

CHAPTER VII

{c|c

We

encircles

orient

c

Since tured by

white

F^,

regions.

tracer h a v e by

has

regions,

For

example,

this a c c o u n t = 6.

f o r m u l a of

P r o p o s i t i o n 7.2.

s ince a

Note

bounds

a disk

Ex er c i s e . the g e n e r a l

in

of

all n ot e

k

is

in

fi g u r e

—1

that

this

In fact,

denotes

that

the n u m b e r

an d is

the b o u n d e d

on p.

of

185,

we

p ( F ^ x) = 7.

in a c c o r d w i t h

H^(F^)

has

We h a v e a d d e d homology

of

k

to o b t a i n a b a s i s tracer

circuits

for

the

as b a s i s c^

c y c le s )

F^.

E x p l a i n h ow c ase

(~

r a nk

where the

v

the p l a n e p u n c ­

c o un t

{c ^ ,c ^ , , c ^ ,c*-,c ^ ,c ^ }.

c^+c^+c^+c^ ~ a

orientation

that

H^(F^),

=p (

p(F„) K.

cycles

see

to o b t a i n

Therefore

the

we

= r an k H ^ ( F ^ / ) - k

ci r cles.

the d ia g r a m } .

type of

In c o un ting,

Then

in

the p l a n a r

the h o m o t o p y

regions)-l.

r an k ^ ( F ^ )

region

compatibly with

the w h i t e

= #(white

a white

^j ( F ^ ) ,•* • ,

in •

and

CHAPTER VII

G iv e or

a procedure

189

for d e c i d i n g w h i c h w h i t e

cycles

to r e t a i n

t h r ow away.

SEIFERT PAIRING We n o w d e f i n e embedding

of a n o r i e n t e d

an d a c y c l e ing

a

normal

0(a,b)

on

0 =

to

: H^( F )

lk(a It

,b). is a n

the e m b e d d i n g Seifert

used

it

to

F,

small

direction

p ai r i n g . of

a

a very

pairing

an a l g e b r a i c m e t h o d

let

a

amount F.

T h is

S -F

this,

.

Given

the

3

---» Z

the

invariant

of

3

,

of p u s h -

the p o s i t i v e

we d e f i n e

by

the

F C S

result

along

the S e i f e r t

formula

is a w e l l - d e f i n e d , the a m b i e n t

bilinear isotopy

cla s s

F C S'*.

invented a version

to be

d enote

into

Using

x H ^(F)

F C S

x

investigate branched

s inc e p r o v e d

of

this p a i r i n g

covering

k n ot

in [ S ] .

spaces.

extraordinarily useful

a nd h i g h e r - d i m e n s i o n a l

Example

surface

for m e a s u r i n g 3

theory.

7 .k:

G

a

b

a

-1

1

b

0

-1

in b o t h

He

It has c lassical

190

CHAPTER VII

The

surface

points

out

F of

is o r i e n t e d the page,

linking

0(a,a)

parallel

c op y

be

computed

=

of

lk(a a

so

towa r d ,a),

along

that the

the p o s i t i v e n o r m a l reader.

a

m a y be

the

surface.

fro m a d i s k w i t h bands,

For

the

self-

r e p r e s e n t e d by a T h us

by c o u n t i n g

0(a,a)

can

c ur l s

w ith

sign.

Example

7.5:

0

a

b

a

-1

b

0

Note:

0

a+b

b

1

a+b

0

1

0

b

0

0

0(a + b ,a + b ) = 0(a ,a ) +0(b ,a ) + 0 ( b ,b ) = 0

T hu s

these p a i r i n g s

embeddings

are

are

isotopic:

6'

c

d

c

0

1

d

0

0

is o mo r ph i c.

In fact,

these

two

CHAPTER VII

191

r?n ~ / ° n ~ ~ fir*■>

We entirely

can, in

if we wan t topological

to do

it,

script.

indicate

a banded

surface

Thu s

represents

the

surface:

192

CHAPTER VII

Exerc is e . face

Determine

the S e i f e r t

pairing

for

this

sur­

F.

SEIFERT PAIRING FOR THE SEIFERT SURFACE Now Seifert

let's w o r k out an a l g o r i t h m pairing

f rom a S e i f e r t

into b a n d - f o r m ) . white in

cycles.

F ^ , .)

diagram

Here

ing

Thus

we m u s t

is a p o s i t i v e

sponding

these

the c y c l e s their

0 (a,b)

c r os s in g .

a

Then

is g e n e r a t e d by

b

for

Th e

labelled and in ord er

drawn.

to c o n t i n u e

regions.

the

contribution

local

Let's of

in

the

shaded,

cycles

the w h i t e

the

6 (a,b).

linking number

surface

it

regions

how each crossing

labelled.

are

pushing

encircling white

wi th Sei fert

intersect

around

0 (b,a)

H^ ( F ^ )

the S e i f e r t

and

the

(w i t h o u t

determine

regions

m us t

courses

and

to

computing

surface

circles

c r o ss i ng ,

regions to

that

( Th e se ar e

contributes

and white

that

Recall

for

corre­ Note follow­ wr ite this

193

CHAPTER VII

Note

that

number

of

negative

a*b

= +1

cycles

on

0 ( a ,b)

=

0 ( b ,a)

= 0.

also, the

+1

where

x*y

surface.

(The

denotes signs

intersection

rev erse

for a

crossing.) Mr

T he

self-linking

(Note:

The

contribution

c yc l e s

is

0(a,b)

bounding white

c o m p a t i b l y w i t h an o r i e n t a t i o n

=

regions

for

= 0(b,b). are all

the w h i t e

oriented

region

itself.)

0 ( a ,b)

= +1

0 ( b ,a)

= 0

0 ( a ,a)

= 0 ( b ,b)

= -1/ 2

0 ( a ,b ) = 0 ^ 0 ( b ,a)

= -1

0 ( a ,a) = 0 ( b ,b ) = +1/2

194

CHAPTER VII

For

e xample:

Here

a

look at

and two

Exerc is e .

b

interact

crossings

Compute

at

on l y one

to c o m p u t e

the S e i f e r t

c ro s si n g.

0 (a,b)

pairing

0 (b,b).

a nd

for

But we

K

of

Figure

on

the

7.1.

Exercise. face F.

Let

Show

x*y that

denote for all

intersection number

sur­

x, y € H^(F),

0 ( x ,y )- 0 (y,x) = x •y .

Hint: general

Do

for S e i f e r t

case.

following be

it

To do

description

two d i s j o i n t

surface bounding t r a n s v e r s a l ly.

surface

the g e n e r a l of

oriented

first. case

Then

it h e l p s

l i nk i n g numbers-' curves.

p.

I so t o p e

Then

lk(a,/3)

a

Let so

= a*B.

B that

Let

try

the

to h a v e a ,p C S

the 3

be an o r i e n t e d a

intersects

B

CHAPTER VII

195

e oc* B = -±.~

> [Why

is

this

E xerc is e . that

this

independent

Prove,

of

the c h o i c e

using Seifert

description

of

(or

linking

of

B?]

spanning)

i mp l ie s

our

s urfaces, original

description. Now

return

to

the

formula

0 ( x ,y ) - 0 ( y ,x)

= x*y,

c o n t e m p late

0 ( x , y ) - 0 (y,x) = l k ( x * , y ) - l k ( y * , x ) = lk(y,x*)-lk(y*,x) =

lk(y,x*)-lk(y,xx )

= l k ( y ,x - x x ) = y- B = x*y.

CHAPTER VII

196

D I F F E R E N T S U R F A C E S FO R A given knot s ur f aces.

For

link can h a v e m a n y d i f f e r e n t

exam p le ,

ra th e r

different

f er e nt

surfaces

two

Seifert

isotopic

surfaces.

s p a n n i n g a k n ot

Th e a n s w e r Consider

or

ISOTOPIC KNOTS

is,

f o l l o w i n g wa y

1)

Cut

out

2)

Take a

will

H o w ar e all

related

in p r i n c i p l e ,

the

diagrams

spanning have

the d i f ­

to one a n o t h e r ?

surprisingly

to c o m p l i c a t e

simple.

a spanning

sur-

f ace : two discs, tube

jointly

S

fro m

ary attached Th i s

is c a l l e d

doing a

1

.

x I

an d

the

surface,

to

dD^

embed

it

but

with

and

1-surgery

to

s uc h f rom

reverse

the

that F

a and

bounds cap

off

consists a disk w ith

two

the

3

d is -

tube b o u n d ­

surface.

F

operation

S

^ 2'

F

Th e

in

af ter

surgery

in f i n d i n g a c u r v e 3

S -F. D

2

‘s.

Then

cut

out

a a

on x I

F

197

CHAPTER VII

X i

af ter Thi s

is a

reduces

O-surgerv.

It

sii p l i f i e s

surface

(i.e.,

g enus).

These

two

surgery

with

the

same b o u n d a r y

faces

the

D E F I N I T I O N 7.6.

Let

opera

F

ions g i v e us d i f f e r e n t

and

F'

be

oriented

sur-

surfaces

with

3

boundary are F

that

are

embedded (F g F')

S-eguivalent

by a c o m b i n a tions

of

S

in if

.

We

F'

say

may

O-surgery,

F

that

be

F'

and

obtained

from

1-surgery and ambient

isotopy.

T H E O R E M 7.7

[LI].

Let

F

and

F'

be

connected,

oriented 3

spanning

surfaces

for a m b i e n t

and

are

Then

F

P ro o f

sketch :

F'

i so topic

L,

links

L'

C S

S-equivalent.

3 » S

3

is

a

: S*

M = (FxO) face where

»

U a(S^xI) in

an d

suppose

from

that

L = a(S

get a n e m b e d d i n g

X, U

i x I

isotopy

T h e n we

x I

embedded W

X = S

the a m b i e n t

L / = a(S^xl). via

Let

.

1

xO)

(F*xl), then

is a 3 - m a n i f o l d

On e

then

embedded

this

is a

s hows in

S

that 3

x I

to

of an a n n u l u s

a( A , t ) = ( a (X , t),t).

3 S x I.

a:S

x I.

If we

in

f orm

closed

sur-

M = dW W

ca n be

X

CHAPTER VII

198

arranged points to

of

so

that

2

type

It m a y be

equivalences

is o b t a i n e d

Now surfaces. tube.

2

x +y -z

the O - s u r g e r i e s

Re m a r k :

a

(S xt) fl W

a nd

of

2

has

2

or

2

-z -y +z

1-surgeries

i n t er e st

between Seifert

on l y M o r s e

to

2

.

critical

These

correspond

we d e s c r i b e d

e arlier.

look d i r e c t l y at

surfaces

for d i a g r a m s

the

S-

that are

from

consider Suppose Then

the S e i f e r t that

H ^ (F ' )

F'

= H^(F)

pairings

for

is o b t a i n e d © Z © Z

S-equivalent

f rom

where

F these

by a d d i n g two

199

CHAPTER VII

extra

factors

are

a nd an e l e m e n t so

that

a*b

We

then h a v e

g e n e r a t e d by a m e r i d i a n

b

that p a s s e s

where

0(a,a)

Because

pairing

= 0,0(a,b) for all

for

of

the

row

An e n l a r g e m e n t Mo r e g e n e r a l l y , S-equivalent tion of pos e b as i s of

of

P

change.)

we w r i t e

and

the

tube

a,

oriented

I

0

0

0

1

P

0 0.

can be

0

and

obtained

Z.

enlargements

and

0 g \p.

If

0

^ from

where

over

as above.

Let 0Qd e n o t e

a_

e0

0

0

0

1

b| P

0

n

0 = a

on c h a n g e

of b a s i s

'

is c a l l e d an

(0 — » P 0 P'

i nv ertible and

b ec omes

0

this k i n d

and

is a c o l u m n v ector.

a 0

two m a t r i c e s

if

enlargement)

lent,

of

congruence P,

tube

0(b,a) = 0

T h e n we h a v e

(0,0,1),

’0 O

= 1, x € H^F).

F.

is a row vector,

p

the

= 1.

0 (a ,x ) = 0 ( x ,a ) = 0

the S e i f e r t

once a l o n g

for

Th i s

S-equivalence. are 0 P'

said

by a c o m b i n a ­ is

the

trans­

corresponds

contractions

an d

to be

ar e

to

(reverse

S-equiva­

200

CHAPTER VII

COROLLARY or

links w i t h

F' \p

7.8.

be

the S e i f e r t

spanning

0

Let

K'

and

connected

K'}.

(for

K

Let

be

pairing

be a m b i e n t F

surfaces

F'.

0

Then

K)

(for

pairing

the S e i f e r t

for

isotopic knots an d

for F

and

\p

a nd are

S -e q u i v a l e n t .

I N V A R I A N T S OF

DEFINITION the knot

S'EQUIVALENCE

7.9.

or

Let K

link

F

be a c o n n e c t e d 0

and

spanning

the S e i f e r t

pairing

surface

for

F.

for

Define (i)

The d e t e r m i n a n t D

(ii)

denotes

(iii)

the

The

= D(t

formula of

= S i g n (0 + 0')

signature

= D(0+0')

of

K,

func t i on o f

signature

cr( K)

D(K)

w he r e

determinant.

The y o t e n t ial by

K,

of

K,

€ Z[t

* 0 - t 0 7).

cr(K) € Z, Sign

where

\t]

by denotes

the

this matrix.

(See d e f i n i t i o n b e l o w . )

Of not g o i n g be

c o u r s e the g a d g e t s to c h a n g e u n d e r

invariants

of

produced

S-equivalence!

exam p le ,

if

0 =

60

0 a

0------- ^— j— a

1 0 .

Hence

they will

and

D(t

0 o'

— ^---- q — j-

.a

V

d e f i n i t i o n are

K.

■0 For

in this

then

0 + 0'

=

0 0.

0 - t 0 7) = D(t

0 Q -t 0 ^)

because

For over

Z

(the

the

s i g na t ur e ,

201

recall

M

c a n be d i a g o n a l i z e d

rationals)

or over

positive

d ia gonal

diagonal

ent r ie s .

the

CHAPTER VII

formula

congruence

The

Sign(M) cl a s s

of

Sign|^

= 0.

is an

invariant

of

invariant

of

invariant

of

K.

and

e+ e_

= e + ~e_. (See

From

its We

Let

denote

Sign(M),

It

is an

[ H N K ] .)

this

shall

also

show

of

is d e f i n e d by

Note

of

the

in p a r t i c u l a r , that

class, that

Q

of n e g a t i v e

invariant

it fo l l o w s

S-equivalence

over

the n u m b e r

the n u m b e r

sig n at u re ,

M.

that

through congruence

IR.

e ntries,

that a s y m m e t r i c m a t r i x

S i g n ( 0 + 0 /)

hence

cr(K)

an

is an

concordance.

T h e po tent ial

f un c t i on p r o v i d e s

a m o d e 1 for

the C o n w a y

p o 1v n o m i a l :

T H E O R E M 7.10. (i)

If

K

links, (ii) (iii)

If

then

K ~ 0,

If links then

K'

and

are a m b i e n t

0^(t)

K,

K

an d

= (t-t

oriented

= 0^/(t). 0 K (t)

then

iso topic

L

= 1. are

r e l a t e d as below,

202

CHAPTER VII

We h a v e a l r e a d y p r o v e d

P r oo f : K.

= 0

if

disjoint connect face

If

K

is a split

spanning

surfaces

these by a

tube

(i) a n d

link. for

To

see

(ii).

Note

this,

two p i e c e s

of

to form a c o n n e c t e d

that

choose the

link,

spanning

a nd

sur­

F.

a

is a m e r i d i a n of

this

type,

then

H^F) S Hj (Fj) ffl H1(F2) ffi Z where

a

= 0(x,a)

generates = 0

We us e surfaces

for

the e x t r a

V x € H^(F), this K,

it

d i s c u s s i o n as K

an d

L.

co p y

of

f ol l o w s follows. Lo c al l y,

Z.

Since

that

0(ct,x)

fi^(t) = 0.

Consider they a p p e a r

Seifert as

CHAPTER VII

We

see

that

H^ ( F ^ )

homology generator L

is a split

and =

Fg,

see

L

may

But

*)

substitution

this

case

= 0

twist.

a

that

while

T hu s

then

= ^

a/_

K.

Hence

determinant

the e x t r a g e n e r ­

F„ a n d 0^ =

with appropriatechoice

Q

on

Frr. K

it

of bases.

calculation

z = t-l/t. the

f o l l ow s

that

f u n c t i o n are

Th u s

reverse.

fi^(t)

It is

to sho w

that

the C o n w a y

related

= v^(t-l/t).

by

the It

is

Then

t = z + 1/t . H en c e t = z+1________. z+1_____ z+ • • • Using

the n o t a t i o n

z+1 , z+1 z+ • • •

we h a v e

[z+y-^ v x^(z) K

for

the c o n t i n u e d

= fiT,( f z + 1 ]->* ). K k J

vKd) =

We

^---- ,

m

our p o t e n t i a l

to sol ve

on

—

our a x i o m a t i c s , an d

should happen

diag r am ,

= 0(a,a) + l.

now a straightforward

polynomial

one m o r e

(iii).

-g— j,

By

have

it

2ir

is not a split

= (t-t

will

unless in

be r e p r e s e n t e d as

0^- =

amusing

F^,

i s o t o p i c by a

that B f a ' . a ' )

Rema r k :

^ ( F^ - )

than

d i ag r am .

proving If

a tor

ar e

an d

203

fraction

In p a r t i c u l a r ,

204

We

CHAPTER VII

shall

return

Let

Example:

to

T

this

subject!

be a

trefoil

with

nT = D

= Lf"o1 - l*.1

0

= (t-t

*)2 +l

Then

= z 2 +l.

-t T hi s

agrees

G i v e n a k no t

Example: the

tan g le

opposite

Since that

in

w i t h our p r e v i o u s

o b t a i n e d by

orientation.

K

K,

= (t

*-t)(-lk(K))

this

cas e

pute

using

with

the

K

Note

K

is a

surface

that

an d h e n c e

exercise

v

that

AND G^(t)

pa i ri n g. of C h a p t e r

the n u m e r a t o r co p y

for

K.

is m u c h

( C om p a r e IV of

*0-t0').

with

we

see

Therefore Apparently,

easier

to c o m ­

this d i s c u s s i o n

these n o t e s . )

Q. = D(t

K

is an a n n u lu s ,

= lk(K)z.

K

of

of

two c o m p o n e n t s .

matrix

the C o n w a y p o l y n o m i a l

last

d enote

link of

is a S e i f e r t

the S e i f e r t

TRANSLATING

let

running a parallel

has a s p a n n i n g

0 = [-lk(K)]

calculations.

Therefore

CHAPTER VII

205

nK (t- 1 ) = D (t 0- 1- ^ 0') = D ( t 0' -t - 1 0) •'• Since

0

is

ponent

links,

where

p.

2gx2g

and

of

for knots,

we c o n c l u d e

is the

To o b t a i n K

= D ( - ( t _10 - t 0 /)).

number

v*,, K

we n e e d . Look

of c o m p o n e n t s

at

2 - 2

.3 -3 t-t

E x e r c i s e . Let

T

for 2 - c o m ­

*) = ( - l ) P + ^Q^(t)

to w r i t e

t +t

T

that

a practical method

z = t-t

that

(2g+l}x(2g+l)

of

K.

oft r a n s l a t i o n b e t w e e n tn + ( - l ) n t n = T v J n

in

terms

the pat t er n :

,

= (t-t

- 1)N2+2

1

. ^- .3 0 0 = (t-t ) + 3 1-31

2

= z +2

-1

3, „ = z+3z.

= tn + ( - l ) n t n and z = t-t *. Show n---------------------------------------- -----

0 = zT .-+T n+2 n+1 n

for

t-t

n

> 0.

* =z

t2 + t -2 = z 2 +2 t3 - t -3 =

z3 +3z

4 _ 4 4 0 t^+t ^ = z +4z +2 5 -5 5 K 3 p. t-t = z +5z +5z

t6+ t-6 Show

that

We for

the

knot.

the c o e f f i c i e n t

can us e

this

second Conway Then

K

= z6+6z4+9z2+2. of

exercise

z^

t^n +t

is

to o b t a i n a c u r i o u s

coefficient

has p o t e n t i a l

in

a^(K).

function

For

in the

let form

n^.

formula K

be a

206

CHAPTER VII

n K (t)

= b 0 + b 1 ( t 2 + t " 2 ) + b 2 ( t 4 - t _ 4 ) + - * - + b n ( t 2 n + t - 2 n ).

f ol l o w s

from our

exercise

a 2 (K)

Exerc is e .

Compute

f u n c t i o n an d

it

that

= b 1+ 4 b 2 + 9 b 3 + 1 6 b 4 + * • -+n2 b n .

Seifert

s ign ature

for

p ai ring, the

determinant,

torus k n o t s

and

potential

links

of

( 2 ,n ) .

type

E xe r c i s e . an d

» K'

show

that

P r ov e is

that

tf(K’) = -cr(K)

its m i r r o r T =

image.

and

when

Calculate

T' = & >

K

is a kno t