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

Table of contents :
CONTENTS
PREFACE
I. INTRODUCTION
II. LINKING NUMBERS AND REIDEMEISTER MOVES
III. THE CONWAY POLYNOMIAL
IV. EXAMPLES AND SKEIN THEORY
V. DETECTING SLICES AND RIBBONS, A FIRST PASS
VI. MISCELLANY
1 Quaternions and Belt Trick
2 Rope Trick
3 Topological Script
4 Calculi
5 Infinite Forms
6 Quandles
7 Topology of DNA
8 Knots Are Decorated Fibonacci Trees
9 Alhambra Mosaic
10 Odd Knot
11 Pilar's Family Tree
12 The Untwisted Double of the Double of the Figure Eight Knot
13 Applied Script—A Ribbon Surface
14 Kirkhoff’s Matrix Tree Theorem
15 States and Trails
16 The Map Theorem
17 The Mobius Band
18 The Generalized Polynomial
19 The Generalized Polynomial and Regular Isotopy
20 Twisted Bands
VII. SPANNING SURFACES AND SEIFERT PAIRING
VIII. RIBBONS AND SLICES
IX. ALEXANDER POLYNOMIAL AND BRANCHED COVERINGS
X. ALEXANDER POLYNOMIAL AND ARF INVARIANT
XI. FREE DIFFERENTIAL CALCULUS
XII. CYCLIC BRANCHED COVERINGS
XIII. SIGNATURE THEOREMS
XIV. G-SIGNATURE THEOREM FOR FOUR-MANIFOLDS
XV. SIGNATURE OF CYCLIC BRANCHED COVERINGS
XVI. AN INVARIANT FOR COVERINGS
XVII. SLICE KNOTS
XVIII. CALCULATING σr FOR GENERALIZED STEVEDORE'S KNOTS
XIX. SINGULARITIES KNOTS AND BRIESKORN VARIETIES
APPENDIX: Generalized Polynomials and a States Model for the Jones Polynomial
TABLES: Knot Tables and the L-Polynomial
REFERENCES

Citation preview

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