Formal Knot Theory (Mathematical Notes, No. 30)

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CONTENTS

lUI L

Introduction •..•...•...•.••....•.••.••....•....••••

:L

2.

States, Trails, and the Clock Theorem •...•.•......•

11

3.

State Polynomials and the Duality Conjecture ...•..•

4.

Knots and Links .....•................•..•••.....••.

5' 67

5. 6.

Axiomatic Link Calculations .......•.•...•..•.•...•.

78

Curliness and the Alexander Polynomial .....••....••

95

7.

The Coat of Many Colors ••....•..•.•..........•.....

105

8.

Spanning Surfaces ...........•...........•........•.

114

9.

The Genus of Alternative Links .....•...•.•....•.•..

125

10. Ribbon Knots and the Arf Invariant ................•.

143

Appendix. The Classical Alexander Polynomial ........•..

156

References. • • • • . . . . . . . • . • . • . . • . • . . • • . • . • • . • . • • • . . . . • • . .

165

1

1.

Introduction These notes constitute an exploration in comblnatortol

and knot theory.

Knots and links in three-dimensional

may be understood through their planar projections.

.,ao.

A knot

is usually drawn as a schematic snapshot, with crossings indicated by broken line segments. trefoil knot. diagram.

ThUS~

represents the

We shall refer to such a picture &s & ~

The projection corresponding to .uch & diagram

forms a (directed) multi-graph in the plane, with tour ed,e. incident to each vertex.

We shall study these graphs .epa-

rately, and for their own sake. In order to do this work it is very important to underline key concepts by adopting terminology and conventions that are easy to remember.

For this reason I have taken a

perhaps startling. but certainly memorable, set of terms for the graph - theoretic side of the ledger. A

(directed) planar graph with four edges incident to

each vertex will be termed a universe. trefoil universe.

Thus

~

18 a

These universes have singularities (the

crossings); they will a180 have

~, ~~, ~

~, ~nd ~.

A ~ of a universe is an assignment of one vertex in the forms

+ +

~

++

per

2

so that

each~~o~ in

marker.

ThUS'

the graph receives no more than one

~

is a state of the trefoil

*~

universe.

Two regions of the state will be free of markers

(since the number of regions exceeds the number of vertices by two in a connected universe). habited by

These free regions are in-

(*).

the~.

It should be mentioned at once that each universe has states.

1n

In fact,

~ ~

!!!£! in adjacent regions

one-to-one correspondence with Jordan trails

~.

~

~ ~ ~­

A Jordan trail is an (unor1ented) path that traverses

every edge of the univlrse oncl and forms a simple closed curve in thl plane.

This correspondence is obtained by re-

garding each state marker al an instruction to split its crossing according to the tollowing schema:

Xr-+>7' /

4

Note that in a state transposition both state markers rotate by one.quarter turn in the same clock-direction.

A state

transposition in which the markers turn clockwise (counterclockwise) will be termed a clockwise (counter-clockwise) move.

A .tat. is said to be clocked if it admits only clock-

wise mov•• ,

~

it it admits both clockwise and counter-

clockwi" MOve., and counter-clocked if it admits only counterclockWi •• MOV••• Tbl kl, oombinatorial result in our study is the followin, ualOl'lm.

ZbI ~

Theorem (~).

Let U be a universe and S the

'1' of .tates ot U for a given chOice ot adjacent fixed .tal".

Then

$ has a unique clocked state and a unique

oounter-clocked state.

Any state in 0 can be reached from

the aloaked (aounter-aloaked) .tate by a series of clockwise (aounter-olookWi •• ) move..

Hence any two states in S are

connected by a •• rie. ot .tate transpositions. By defining 8

< 8'

whenever there is a series of clock-

wise moves connecting the state 8'

to the state 8, the

collection of states becomes a lattice whose top is the clocked state, and whose bottom is the counter-clocked state. This result is illustrated for a particular universe in Figure 1.

> 1t*

Let F: ft

the underlying string structure.

be the

mapping that associates to a string with sites its corresponding string, and

p-l: 1£*

->

i

the inverse map.

With this

understanding, we have an immediate extension of the derivative to D:

~ ->

the right hand

it

via the formula

D(X) = p-1DF (X)

D our original derivative on strings.

with The

higher derivatives now exist via iteration of this operation. Lemma 2.16.

Let A be a string.

tive integer N such that DrlA

Then there exists a posi-

= riA.

Let

DA

denote this

Itring with sites, and call it the dissection of !.

Then

the dissection of a string A is a shell composition with Iltes. l~ot.

This to1lows easily from the definitions.

In a composition ot shells the vertices on each shell can be split in either clockwise or counterclockwise fashion," as illustrated below:

"30

CX

(clockwise)

C'X

(counterclockwise)

If Y is a composition of shelll, we let

CY

denote the

result of performing a clockw11e Iplit on each shell (similarly for

C'Y) •

Theorem 2.17.

K'A

= CIDA.

Let A be & .trln,. Then the tra11.

KA

Let

and

K'A

...

KA = CDA

and

correspond to

clocked and counterolooke4 .t.te. of A respectively. Thl1 theorem oompletel cur statement of the algorithm tor oonltruotlng extremal Itates.

In order to prove it we

Ihall analyze the sort of sites that may be added to a shell composition to obtain a string with sites of the form By specifying allowed sites the proof will emerge.

...

DA.

Figure 7

illustrates this algorithm as it applies to the example just prior to Definition 2.9. Allowed and Forbidden Interactions It is often convenient. when viewing a string with sites to see it decomposed into the same atoms as the pure string obtained by closing all the sites. the map

. J: ft - ) 1t

To this end we define

that closes all the sites.

A string

with sites will be said to be J-atomic or J-irreducible when

31

u

DU

Deriving

~

Clocked

Figure 7

~

32 its image under

is atomic or irreducible.

J

A pure string

(without sites) divides the plane into regions (with two unbounded regions by our conventional.

By these same conven-

tions a trail T divides the plane into two regions, but the regions for the corresponding string, J(T), are apparent from the diagram, Since they are bounded by edges and sites. sites correspond to

doorw~.

The

in rooms - one can identify the

interior of a room even when the doors are open.

Therefore A

we define the !22m! at a Itring with sites regions of the pure .tr1ns J(X).

X E~

Since J(J(X»

to be the = J(X), the

regions ot a pure Itrlnl are identical with its rooms (and some roOllls

m~

haVe no doors).

Conlider a oomposition ot ehells Y and a single circle a

in Y.

Then

lower arc a_, and q Let

00

a

11 divided into an upper arc Q+

Here

a-a+Ua_,o+no_=(p,qj

are the intersection points of denote the intersection of

(bounded) region determined by a, and 00 -may each have riders. U1'U2 ••• · ,Uri 0_

A

°

and a

where

with a line

p t.

with the interior

In Y. the lines

Suppose that

0+

0+, 0_

has riders

Ll •••• ,Ls; a o has riders Ml ••••• Mt • Each rider is itselt a shell composition, and the riders are all disjoint except for their connections along 0+, 0_

°

has riders

and

and its riders.

°0 ,

Fo denote this composition of See Figure 8. Let

33

Figure 8 Let

Fa

plus its

and Fa

= riders,

denote 0

respectively.

a=

plus its riders, and

Thus

Fa = (input and output lines) U Fa U Fa U Fa . Ipecify how sites may be lites

F~, so that

ao

We wish to

+ 0 added to Fa' torming a string with

DJ(F~)

= F~. A point

of

p

a.

FI

will

be said to be a pure boundary point it there exist paths, confined to single rooms of output strands of of

F~,

F~,

from

p

to the input and

and also a path from

p

to some point

Using this concept, the following interaction rules o are adopted: F~.

Interaction rules. 1.

•

Let

denote a string with sites •

Then - ......-

may be replaced with either ot

the two torms: 2.

It .0. FI

is a string with sites that has been obtained

trom a shell composition ot the torm Fa' then further sites may be added between

and F~ as long as o :I:: the point (cusp point) of the site com1ng trom F' F~

Cl.:l:

is a pure boundary pOint. }.

Rules 1. and 2. may be app11ed to any J-rider on a liven 'trins with 'ite,.

With "Iard to

rul. ", note that a J-rider 1s simply a sub-

.,.1ns wlth 11t•• that ha. no .1te interactions with its canta1ft1nl .tr1n. (h.nce it corre.ponds to a decomposable part under

J,).

Propos1tion 2.18.

Let

SH

denote all strings with sites ob-

tained by elaborating shell compositions according to the ~.

interaction rules 1., 2., and 3.

Let

t10n of s1 te - free strings, and

J: SH

denote the collec-

->

~

the mapping

that closes all sites. J

0

Then J

is a one-to-one correspondence, "and D 0 J

D=~.

Thus

SH

= lSH' .

is exactly the set of elaborated shell

compositions obtained by dissecting strings via

D.

35 Using the Interaction Rules Here are a tew examples of the application of the in' •• action rules:

~)')F'

_, 1.

.-/

't:::Y' '--

!lJF' = F'

~ boundary point G

l-..;:;.:...2._~

G'

roo'

.a

G'

Proof' of'~.

D.

J = lSH'

Since

J

D = ltt'

0

it suff'ices to show that

Since this property is preserved under applica

t10n of' the interaction rules, the result follows by induetion. ~.

It is worth observing how things go wrong when the

interaction rules are violated. site to J(F') =

--&

in

~

DJ(Ff) ! F'

(and

DJ(F')

-

=

-@-.

Then Thus

violates rule 2.).

Before proving Theorem placement ot atate

~'.

the f'orm

and F'

For example, if we add a

ma~ke~l.

2.17.

we need a lemma about the

Conlider an atomic string.

g!ft

Take ita tirat derivative and start putt1ng 1n the markers f'or the clocked state that will result from the algorithm of' Theorem 2.17.

Notice that markers at sites between the top of the shell (the shell produced by taking the derivative) and the middle .tring go to tbe right in tbe 'oro

~

where the +

sign labels the cusp from the top part

ot tbe shell.

• ......

larly, the markers tor sites between the middle and the "',.. part ot the shell point to the left.

This phencmenon ,ropa-

gates, and we obtain the Lemma 2.19.

Let

X E SH

be a sheU composition with s1te.

al.lowed by the interaction rules. clocking all shells in

Let ex, the result ot

C. be decorated with state markers·

according to the procedure ot Theorem 2.4 (states trails correspondence).

Let Fa

be a shell configuration within

X, as depicted in Figure 8.

Denote a site with contributing

and a site and Fa by the notation ~ ~ + 0 V with contributing cusps trom Fa and Fa by A o Then markers tor these sites will appear in ex on the

cusps from Fa

right and lett, respectively.

That is, they will have the

y

y

foms

Proot of 2.19.

A

and

A

Combine the observation just prior to the

statement of this lemma with Proposition 2.18. ~ .2!~.

the trail

cbA

Let A be a string.

= KA

We wish to show that

corresponds to a clocked state.

it is easy to see that

ex

Since

is clocked whenever X 1s a pure

shell composition (without sites), it will suftice (by 2.18) to show that if XI action rules, and clocked.

is obtained trom X E SH via the inter-

ex

is clocked, then

ex'

is also

We must show that the addition of an allowed site

38

cannot create a counterclockwise move. We can limit our considerations to interactions on a form Fa as depicted in Figure. 8. figure.

Terminology will refer to this

Call Fa the top, Fa the middle, and Fa_ the + 0 Then a new site may be between middle and top,

~.

middle and bottom, or it may be a self-interaction of one of these forms.

Using the notation of Lemma 2.19, we may assume

that all middle - top or middle - bottom sites receive markers in the forms

~

A

~ ~

and

By interaction rule 2 ••

the cusps from the top and from the bottom form parts of·the boundary between the interior and the exterior of the curve Q,

Theretore & counterclockwi.e move involving either of

tho •• markerl would nlce"arlly move the marker into the ext.r10r ot

oan

Q,

Therlfore no interior counterclockwise moves

arl,. trom '1t., ot thi. type. Pina11Yt con'ider the introduction of a top-top, bottom-

~ottomt

or middle-middle site.

If this occurs along Fa • o

-e-

then it will have the form

A counterclockwise move then entails an interaction of the form

y

Y ~

~-.~Since the marker at the

or (x,I3)

A

site is on the left and

13

is part of the middle portion, x must also be part of the

39 middle portion (since top - middle sites havemarkeZ'. right).

But, this is a forbidden interaction.

Eaoh t1m. a

self-interaction occurs, a curve or composition off.

Further self-interactions must come from

on th.

~ ~

18 .,llt

itselt.

Thus no counterclockwise moves can come from self-interaction. of the middle. Similar arguments apply to the top and bottom. have shown that

CF

Thus we

admits only clockwise internal moves.

Since any composition in

SH

can be decomposed into forms

of this type, we have shown that counterclockwise moves.

KA

does not admit any

It remains to prove the existence of

clockwise moves in KA. As we observed after Lemma 2.16, a composition of curls has no available moves.

This may occur when the shell com-

position underlying

is trivial.

DA

composition is non-triVial then move.

KA

If, however, this shell does admit a clockwise

Such moves can be located by searching for a deepest

!h!!! in

DA.

A deepest shell in a shell composition is a

shell whose middle, top, and bottom have no riders.

Such

shells exist since there are a finite number of shells in the composition.

It is then easy to see that an elaboration of a

deepest shell Via the interaction rules will always admit a clockwise move. Remark.

This completes the proof.

Figure 9 illustrates some elaborations of riderless

shells, and the available clockwise moves.

40

Fi~re

9

We are now readr to approach the proct ot the Clock Theorem.

The cor. ot the proot rests on a procedure for

loina trom one .tate to anf other state bf a series of transposition..

Once this methOd i8 clear, the theorem will

tollow, and we will be able to show that the extremal states constructed by Theorem 2.17 are unique.

In order to·create

a series of transpositions between two states, we first show that there is a series ot exchanges between any two trails. Each exchange then factorizes into a series ot clockwise or counterclockwise moves between the corresponding states. Definition 2.20.

A trail T'

trail T by an exchange it T' two sites of T.

is said to be obtained from a is the result of reassembling

41 Note that upon reassembling a single site, a tratl will break up into two components.

If these two componente

in'.r-

act at another Site, then a second reassembly at this Ittl will constitute an exchange.

Using string form, the first

reassembly produces an extra component that is homeomorphio to a circle.

Thus the generic form of the exchange (ignorinl

the presence of other sites) is as illustrated below.

If there are no other sites between the top and middle, or between the middle and bottom, then the exchange 1s accomplished by a single transposition of the corresponding states. With intervening sites, a series of transpositions can do the job (as we shall prove).

Examine Figure 10.

Since the generic

form of the exchange replaces a clocked form with a counterclocked form (or vice-versa), we shall refer to clockwise

~

counterclockwise exchanges where a clockwise exchange replaces a clocked form by a countercloc"ked form.

In Figure 10 we see

that a clockwise exchange corresponds to a series of clockwise transpositions. Proposition 2.21. U. T' •

This is always the case. Let

T and

T'

be trails on a universe

Then there exists a sequence of exchanges taking T to

42

'exchange

Factorizing

~

Exchange

~

Figure 10

Transpositions

Proot

~

2.21.

Two trails on the same

at an even number

01'

u~iverse

mu.t dirt.r

sites (since one trail can be tranl 8

tormed into the other by reassembling all the sites at whiDh they ditter, and an odd number connected torm).

These sites may be paired ott with each

other to give the desired set Proposition 2.22. U so that T' change.

Let

reassemblies leaves a 411-

01'

01'

exchanges.

Let T and T'

be trails on a universe

is obtained trom T by one S and S'

be states

01'

U (With the same

star placement) corresponding to T and T' Then S'

may be ob.tained trom S

transpositions.

clockwise~ex­

respectively.

by a sequence

01'

clockwise

Except tor the state markers at the exchange

sites (which must each turn through 90°), any state marker involved in these transpositions will turn through a total 01'

either 1800

or 3600

•

Thus clockwise (counterclockwise) exchanges factorize into clockwise (counterclockwise) sequences of transpOSitions. Proof'

~

2.22.

We shall prove this result by induetion on

the number of vertices in the universe U.

In order to do

this induction it is necessary to state the details of the factorization procedure more precisely.

The generic f'orm of

the clockwise exchange is that of' a clocked shell.

As such,

it has a top, bottom, and midline as indicated below.

44 midI E€t~ ~

in,

-bottom

In practice, the top, middle and bottom will all have extra sites.

The midline itself is a trail with sites (self-

interactions) and cusps (places where the midline has sites with top and bottom).

Let

~

denote this midline trail with

its sites and cusps.

Then

~

may be written as a sum,

~

= ~l e

~2

e ...e

~n'

of J-irreducible trails with sites and

cusps, where we extend the notion of J-irreducible and J.atomic (see discussion atter 2.17) ( --""'--- or

~)

by

taking an isolated cusp

as J-atomic.

Recall that a

trail is J-atomic it the string obtained by clOSing allot its sites is atomic.

For the rest ot this proof, atomic

(irreducible) will always be used in place of the term Jatomic (J-irreducible). Each

~K

is then a composition of atoms, and the atoms

are partially ordered is a rider on B.

by

the relation:

If A < B and

that atoms on different

~K

B

A < B whenever A

-< .... )

Note that

are the saae as those on

46 U, and that an otherwise identical exchange problem is presented tor U'. Its

The induction hypothesis applies to U'.

is a site on an uninvolved atom, then no new in-

volvement is created by its removal. the tactorization tor U'

Hence, by induction,

extends to a factorization tor U

that still satisties the induction hypothesis. Suppose that

s

is on an involved atom, and that all the

crossings nearest to

s

remain involved when

s

is removed.

Then, by induction, these nearby markers undergo, rotations as described by statements 1. through 5. these rotations in U' at

Upon replacing

360·

induce a

s,

rotation of the marker

The geometry ot this induction is illustrated in

s.

Figure 11. Finally, suppose that when

s

s

is on an involved atom and that

is removed, one or both of the segments from the site

belong to uninvolved atoms in U'.

In this case we are pre-

sented with a situation in the torm

.,,--t¥-...

...--q;r-...

.

8" .: - ... ~ ... \

:

....~ ...

~

1.1.'

...

Ci

The site B.

s

is an interaction site between two atoms

These ride on the larger atom

When

s

involved.

is removed, the atoms

A and.

C, which is involved in U.

A and

B may no longer be

We have illustrated how B could lose involvement:

47 In U' B

a direct transposition at

need not be utilized.

x and y is &v..UI~lI, and

Note that by induction, thil

position,does occur in the factorization for U' larger atom

'rift••

(s1nce 'he

We are now pre.

C is still involved in U').

sented with a small factorization problem of the same typel Namely:

\ - - - -. .1-::::-- ••• ";J

Since this occurs on a smaller universe, the induction hypothesis applies, and we obtain a partial factorization in U where the marker at

s

turns through

180".

Now apply the

same reasoning to A, and obtain the full rotation of 3600 tor the marker at

s.

This completes the induction, and hence the proot of Proposition 2.22.

3600 Induction Figure 11

48 Remark.

Here is a concrete example ot the last part ot the in-

duction argument tor 2.22. Thus

U'

Let U and

UI

be as shown below.

is obtained trom U by deleting the site at

(indicated in state torm).

The atoms

A and

B in U'

uninvolved, but become part ot a larger involved atom U.

s are

C in

Contained within the larger factorization for U are the

small tactorizations with A or B and the site

.__A

u:

··

.

•

s•

49 The clocked state is unique. We are now prepared to show that the clocked stat. il

In

unique (and that the counterclocked state is unique).

It

order to do this, consider the form of an atomic trail.

it is a curl form, then no exchanges are available, and it is the only trail on its universe; thus there is nothing to prove.

Thus we may assume that the trail

an atomic,string A that is not a curl.

T corresponds to This means that

T

can be obtained from the schema

by

1.

splitting the vertices at

x

and

y,

2.

connecting {a,b,c} and (a',br,c') 1. and :2. produce a c'onnected curve,

}.

adding extra sites except at the input and output lines.

so that

In Figure 12 we have enumerated the three basic possibilities for such an atomic string.

Type

a

illustrates a clocked

shell upon which extra sites may be added. include a counterclocked shell type

~

the sites

x

and

y

a'

We should also

under this case.

In

have been split in a clocked

(or counterclocked) manner, but the pitchfork connections have been made in the one other possible manner. trated is

~o'

Also illus-

the form with the least number of sites that

is in this category.

The same remarks apply to

'Y

and

'Yo'

Any trail in the categories

~

extra sites to the prototypes Since

~o

and

~o

.or

~

and

~o

is obtained by adding ~o.

admit both clocked and counterclocked

exchanges, no trail in these categories can correspond to a clocked or counterclocked state (using Proposition 2.22). Thus a clocked atomic trail is in type clocked atomic trail is in type outer form of the trail.

at.

a, and a counterThis determines the

By repeating this criterion on the

resulting midline trail (recursively) we arrive at exactly the description of the clocked state that is summarized in Theorem 2.17. This completes the proof that the extremal states are unique.

Since any state can be transformed into a clocked

state by performing successive counterclockwise moves, this shows that any state is connected to the clocked (counterclocked) state by a sequence of transpositions.

Figure 12

51 The collection of states is a lattice. A specific marker in a state can be involved in no more than one transposition at a time.

Consequently, it 1s

~Ossi­

ble to label all state transpositions trom the clocked .tate on one diagram in the pa.ttern

For example:

In this example the moves labelled stricted by the heirarchy

and

b2

are re-

~

/"

bl

That is, bl

a, bl , b2 , c

b,

\c

cannot be performed before

a, and

c

must be done after and then do

b2 . If we write rs to indicate "do r S", then rs = sr whenever this makes sense. ab2 c # aCb2 since cb2 is Here equality means that~the resulting states

That is, abl b2 = ab2bl meaningless. are identical.

but

ThUS, trom the heirarcby ot operations we

52 sen.rate the collection of states

(Here

1

denotes the

clocked state.)s

Recall that a lattice is a partially ordered set that every pair ot elements

X and

Y of

S

S

sue~

have a well-

defined infimum X ~ Y and a well-defined supremum X V Y. If we indicate states by clockwise move sequences as above, then

X V Y

is the move sequence corresponding to the in-

tersection ot the set of moves for this detines a unique state.).

X and Y (Observe t~i

Similarly, X "Y

to the union of the moves tor X and Y.

corresporia.."

With the partiaJ.: '

order X < Y whenever there is a series ot clockwise movei ' trom Y to X, this gives a lattice structure to the set of states ot a string. The proot ot the Clock Theorem is now complete.

53 ,1.

~

Polynomials

~

the Duality Conjecture

Suppose that labels have been placed in each ot the four corners ot every vertex in a universe kth

% k

D

k

and suppose that the universe has ~

K and a state

K denote

vertex, Vk • are

B

Define the

Let

For purpose of discussion suppo,.

this labelled universe. that the labels at the

U.



product

W k n

vertices

(k. 1, •.•• n).

between the labelled universe

S of this universe by the formula

-

a(S)Vl (S)V2 (S) ••• Vn (s) where Vk(S) is the label touched by the state marker of S at this vertex. when the state and the labelling forms are superimposed.

Vk(S) = ~, Wk' Uk' or state marker of of the state We regard

Dk according to the position of the

S at the given vertex.

a(S)

is the sign

S.



as an element of the polynomial ring R

whose generators are the collection of labels of coefficients).

Thus

K (integer

The state polynom1al for a labelled universe

K is then defined by the formula

The state polynomial, appropriate matrix:

Let

S(n), we see that this propos.ition follows directl, from the definition Det A(K,l')

t

pES(n)

of

the determinant

sgn(p) Ap lAp 2" .Ap n' I 2 n

55 The state polynomial will be specialized to various sorts ,

of labelling.

Tne reader should compare this formal develop-

ment with the discussion in the introduction. The first labelling will be the standard

~

~

which we shall abbreviate as siMply

~

A blank corner indicates the label 1. Proposition~.

Let

oriented universe U label).

K be the standard labelling ot an (i.e., each vertex receives a standard

Let 8 be the set ot states tor U with a given

choice ot adjacent fixed stars.

Then tor a state

S in 8,

= (_l)b(S)Bb(S)WW(S) where b(S) denotes the number of black holes in S, and w(S) holes in S. N(b,w,8)

Hence



=

t

denotes the number of white

(-1)~(b,w,8)Bbww where

b,w is the number of states in 8 with

b black holes

and w white holes. Proof. Remark.

This follows directly trom definitions. Propositions 3.1 and

,.2

taken together show that

the states of a universe may be enumerated by taking the determinant of a matrix.

This is an analog of the well-

known Matrix - Tree Theorem (see [9]).

The Matrix - Tree

Theorem enumerates rooted trees in a graph.

In

tact, the

states of a universe are iri:one-to-one corr~spondence with rooted trees in a graph associated with the universe. iraph, G(U), is obtained as follows:

This

Checkerboard color the

regions of U with the colors black and white (say that the unbounded region receives black).

The vertices ot G(U)

are

in one-to-one correspondence with the white regions of

Two vertice.'in

are cormected by an edge in

G(U)

U. when~

G(U}

ever the corre.ponding white regions share a crossing.

For

example:

It is ea., to ... t:rOlll the proof of Theorem 2.4 that maximal:' rooted 1EUJ. in

A.aU

m s: JiUU..u

1l.

l8e this rl.uU, let

are.!!! one-one correspondenee ~

m.! given .£h2!£!

of

fixed~.

.:l1!!!

To

denote the (dual) graph obtained.

G' (U)

by thl saml con.tJ'\\ction on the black regions.

mal rooted tree on either of the graphs

Then a maxi-

G(U) or G'(U)

uniquely determines a tree on the other once the roots are specified--together the two trees exhaust the set of crossings in U.

A pair of trees specifies a state as in Figure.'t :'"::1":1

The Matrix - Tree Theorem has been applied in knot theorY via the graphs

G(U)

and

G'(U)

(see [8], [9], [12J. [3:H);'2' "

To the best of my knowledge, the cormection between trees in>;\ G(U)

and the classical Alexander matrix has not been noted

betore. The main result in this section is Theorem hl.

Let

K be a standard labelling for an oriented

uni verse

Let

J and J.

U.

be two collections of state.

for different choices of' adjacent fixed stars.

Then

... -W •

(The black holes contribute the minus signs via the formula for the sign of a state S: cr(S)

= (_l)b(S).)

= (W-B) .

Since

Hence -

can-

not see up - states or down - states, -

= (W-B).

This completes the proof

of the exchange identity. We use the exchange identity to prove independence of star

70

placement.

Since

L

has fewer crossings than

may assume by induction that of stars. if



K,

we

is independent of the choice

Hence, by the exchange identity,

is independent.

K or



is independent

Since any two links with the same

underlying universe can be connected by a sequence of crossing exchanges, it suffices to produce one is independent of the star choice. Theorem 3.3 for

K so that



This is provided by

K the standard labelling.

The proof of 4.1

is now complete.

*~ +W~ J .~ -we + 8' R :1..

::1.

A(K) .. ~ 0

.

8

:1. 'W'

A(K~.Lr

/



= BW.

In Case B'we have the cod1ng

Aga1n. states of type

Sl

and

S2

tr1bution to the state polynom1al.

cancel each other's conFrom the code above and

the correspondence of states, it then follows that invariant under the move of type II B. verif1cation that



is

This completes the

VK is 1nvariant under moves of type I.

and II. Analysis of the cases for the moves of type III. will proceed by listing state types for the star placements

75

" . , ' " A ."

..

~

These indicate corresponding regions before and after the

move

(compare with Figure 22). For example, suppose we consider a state of the form.

Here the

•

signs occupy regions that have state markers

other than at this triangle.

The state depicted above is the

only type for the horizontal line in the upper position. ing the horizontal line downward, we find three types

Slid-

So'

~¥~~

In order to catalog the contributions of these states, suppose that

K and KI

(differing by a type III move) are given

locally by the diagrams

K,:

"

Za

X y

/ /"

,,-

=

76 Le~

a

denote the contribution to (Kls> from outside the

(K'IS~=

_w2 a, 2 a. Hence (K' Is~ USi U Sp = a. This (K' lSi> = a, (K' 182> - W given triangle.

Then = BWa while

proves that . - t = VK,. as predicted 6.

a Weave:

Figure 27. Claim:

(W-B) 4 + (WB+2) (W_B)2 + 1.

Let,

1\ f+

,by

RI\

Theorem 4.3.

be the operation indicated in

-n -

Then, as shown in this Figure. R"-".

v

n R "

=

2

"1\- z v n-l R

"

86 ~.

Let "~ • RnA.

Thus Xn

N

A.

~~~ L""

-zvXn-l Q.E.D.

Figure 27 In Figure 27 a knot

A is given with unknown tying within

the black box, and some particular starting configuration. The transform RA

is a knot of the same torm.

The claim

shows that we may recursively calculate the polynomials for these knots .• For example, let

"

be a trefoil knot.

;; is an unknot. and we have (letting fn tn

=1

2

- z f n _l •

= v~)

fo

Then

=1 +

z2,

87

... fO = 1

+ Z2

f

.. 11m f •

n-

=

246 1 - z + z - z :!:".

n

fl = 1 - z2 - z4 f2 = 1 - z2 + z4 + z6

It 1s tempting to try to make sense out ot ing

f.

such that

f.

by assign-

as an extended invariant for an infinite knot

RK.

idea, see [20).

~

K.,

K.

For a possible formalization of this

88 7.

Tangle

.!hee ".

This)~l' 18 a quick introduction to Conway's theory

ot tangles. ways.

The material here can be generalized in various

See [5] and [15J for more inf'ormation.

A tangle is a

piece ot a knot diagram with two input strings and two output strins., oriented as in

Figur~

28.

Each input is connected

to one output, and there may be any other knotting or linking (Without tree ends) inside the tangle box. and

I, the tangle

output.

61

Given tangles A

A + B is defined by connecting inputs to

in Figure 28.

Also, there are two ways to form a

knot or link from a given tangle A. (numerator) and D(A)

These are denoted N(A)

(denominator) as N

Given a tangle A, let A

= VN(A)

in

Figure 28. D

and let A = vD(A)'

The traction of the tangle A is then defined by the formula F(A) _ AN/AD, Theorem!2.d,. + F(B),

Let A and B be tangles.

That is, (A+B)N _ ANBD + ADBN

Then F(A+B) ... F(A)

and

For example, consider the tangles Q and Q=

~:

-.. --;~

~.~

Q+Q=:t=o Q+ ~

(A+B)D = ADrf,

= ::;> c!::

=..

~ + !: ...

::t> c:!::> c:r: = ~l

¥= ¥+ ¥. . F(Q) + F(Q) F(Q+~) = F(~) = ~ = ¥+ ~ = F(Q) + F(~)

:. F(Q+ Q) ...

F(!!+!!) ... F(!!l)"'~ = ~ + ~ ... F(!!) + F(!!).

89

= 1/0 +

Note that the identity % formal addition of fractions

1/0

tollows from

alb + c/d = (ad + bc)/bd.

we have verified the theorem for the tangles .Q and !!.

Thus It

will be left as an exercise for the reader to show that this actually suffices to prove the theorem!

(compare with Lemma

5.2. )

As an application of this addition theorem, let Ln note the link illustrated in Figure 29. Vtn = Num(Z + ~ + ~ +.•• +~)

5.4 that

denotes a continued fraction with n

It then follows from where this formula

terms.

t A+B

N(A)~ tj

B

P

D(B)

geoo N(A)

Figure 28

de-

0 (A)

90

Figure 29 8.

The Coefficients Let

~K(Z) = ao(K) + al(K)Z + ~(K)z2 + a,(K)z3 + •••

Then the coefficients of the polynomial are themselves invariants of link type. that

ao(K)

= {l o

It follows immediately from the axioms

if K is one component otherwise.

For certain calculations it is useful to adopt a notation to indicate the operations of switching and eliminating a crossing.

Accordingly, let

E(X) =

E(X.)

=

S(~)

~(X) = ~ and let EiK

and

operations to the

SiK

='X

denote the result of applying these

ith crossing of K.

We have implicitly, up until now, aSSigned indices to the crossings according to the formulas:

IeX'):: Let

+i

IiK denote the index of the

)

I(~)

=-i.

ith crossing of K.

Then

91 the exchange identity of axiom 3 may be expressed by the formula

It follows that the coefficients of the Conway polynomial obey the series of identiti•• :

In particular, it is now easy to ••• that terpretation in terms of linking Definition~.

al(K)

has an in-

num~.r.1

Let K be a two-component link.

Define

the linking number, Lk(K), by the formula Lk(K) = i tilT(K)Ii(K).

Here the set T(K)

of crossings of different components of K. do not contribute to tne linking number.

Lk((9)) = Lemma~.

Proof.

denotes the set Self-crossings

For example

+ -1- •

a1(K) = {Lk(K) if K has two components, o otherwise.

This follows at once from the definition of the link-

ing number, and from the exchange identity for Remark,

al ,

It is obvious from its definition that the linking

number is invarfant under elementary moves·, The problem of direct interpretation for

~(K)

and for

the higher coefficients of the Conway polynomial is more mysterious! ~(K)

We shall now give an interesting property of

taken modulo-2, and in section 10 we shall show that

a2 (K) (mod-2)

is the Arf invariant «(34]) of the knot

K,

lat ed two kn ots th at are re Le t K and K' be am below: ac co rdi ng to the d1agr

TheOre!!,.~.

oth erw ise 1d en tic al fo r K and K' are Th at 1s , the diagrams (m od ulo -2) . Then ~(K). ~(K') ld cro ss1 ng as in rti ce s at the to ur -fo Pr oo t. La be l the ve dic ate d below. = E4S3S2SlK~ = ~S2S1KJ an d X4 a E2S1K, ~ , ElK l· Le t X pli es th at ca tio n ot axiom :3 im pli ap ted ea rep a en Th cu lar , + Vx - "'X 4 ). In pa rti vK - VK, .. z(vX 1 - Vx 2 :3 a1 (X 4 ). Th is Xl) - al(~) + a l ~( .. ) KI ~( ~(K) of a 2 to a problem of th e change the es uc red la mu for so cia ted lin ks mbers to r the to ur as nu ng ki lin of rn tte pa as an ex erot the pr oo f 1s le tt st re e Th • X X" 4 , Xl ' X2 ms of co nn ec tio n e are thr ee ba sic for er th at. th te No e. cis e component}. ng to form a kn ot {on ssi cro ld -fo ur fo the to r schamas: They co rre sp on d to the

X3

(X3) -

•

93 Remark:

Theorem 5.7 also holds for a four-fol.d ClZ'OIl:Lnl

switch with reversed orientations on one pair strands.

Thus

K and K'

or

~ar&11.1

forml

• In general, all that is required is that each pair of parallel strands have opposite orientation.

Call two knots pass-

equivalent it one can be obtained trom the other by a combination of ambient isotopy and four-fold switches of this type. For example, the trefoil and the figure eight knot are passequivalent:

~ trefoil In section 10, when we discuss the Arf invariant, we shall prove the following theorem. Theorem. only if'

Two 90ts

K and

~(K). ~(K')

K'

are pass-equivalent if and

(modulo-2).

In particular, any knot

is pass-equivalent to either the trefoil knot or the unknot. This theorem provides & direct geometric interpretation for the

mod-2

reduction of

~(K).

94

o

9,

Co ns ist en cy ,

8, le t no tat ion of example Us ing the sw itc hin g . ori~nted lin ks ere K and K' are wh lK "'S _l Sr Sr = K' id en tit y shows ati on of the exchange Then rep ea ted ap p1 1c

so th at K' is un a sw itc hin g seq ue nc e Sin ce we can ch oo se s th at it sh ou ld be th is for mu la su gg est kn ott ed or un lin ke d, ter ms .of an ' In ely de fin e an +l in tiv uc ind to ble ssi po the Conway po lytak e th is ap pro ac h to [2J Ba ll and Mehtah ry pr oo f of the ele ga nt and ele me nta no mi al, ob tai nin g an eir ap pro ach is the iom s. The ke y to th ax the of cy en ist ns co d in Fig ure 24 , pro ce du re ill us tra te ing ott kn un the of e us r pa pe r fo r fu rth er r is re fe rre d to th ei de rea ted es ter in e Th de tai ls.

95 §.

Curliness

~

!n! Alexander Polynomial

In this section we discuss the structure of a univt,.. viewed as a plane curve immersion.

The total tuminl numbe,

of the tangent vector to such an immersion will be given a combinatorial torm called here curliness.

We then show that

Alexander's original algorithm for calculating the Alexander polynomial contains a hidden curliness calculation.

It is

this calculation that make. the Alexander polynomial dependent up to factors' of the form • tk, upon the particular choice of knot projection for a given knot.

Thus we show how to

normalize the classical polynomial to obtain the state polynomial model for the Conway Polynomial. prove the translation formula the Alexander polynomial,

~(t)

In the process, we

= vK(Jt - l/Jt)

relating

~(t),

(Here

=

and the Conway polynomial. denotes equality up to factors of the form ctK

where

k

is an integer.)

Immersions If

ex: 81

->

R~

is a differentiable mapping of the

circle into the plane with non-vanishing differential

da,

then

is

a

is said:to be an immersion.

said to be normal if the pre-image p E R2

ex-l(p}

ex

of any point

is either empty. a Bingle point, ,or two pOints

(Pl,P2) = a-l(p) pendent.

An immersion

with

dex(Pl}

and

da(P2)

linearly inde-

Thus a normal immersion is a locally one-to-one

mapping with normally crossing singularities in the image,. The same remarks and definitions apply when the single circle

96

is replaced with a collection of disjoint quently, ,g, universe !! may

~

c~cles

A.

Conse-

represented .!!. the image

a(A)

!2t..!: ~ immersion a: A - ) R2. where II ll!: diS-

joint

co~ection

of Circles.

a,~:

Two immersions

1\ - )

R2

are said to be regularly

homotopic i t there is a family of immersions

o~ t

~

1

so that to = a, fl =

continuously wi tb t.

and

dtt vary Just as ambient isotopy tor knots and ~

ft

t t : " - ) R2 • and

links is discretized by the three elementary moves (Rei demeister moves) of sections 4 and 5, we obtain a discrete version of regular homotopy via the moves of type A and B illustrated below:

A.

B.

Definition 6.1. homotopic

Two universes

(U .. UI

)

if U'

U and U'

are regul,ar1y

can be obtained from U by a)

sequence of moves of type A and

B.

G1ven an immersion a: 81 - ) R2

of an oriented circle

into the plane, Whitney [}8] defined a degree D(a) E Z.

The

degree measures the total. number of times (counted with sign) that the image unit tangent vector turns through 2".

as the

97

curve is traversed onoe. abo~t

clockwise circle

By convention, we tate & counte,-

the origin to have degree

and Graustein proved the fundamental result:

ex. ~ : Sl - )

R2

are rell-\l&:rl.y homotopic

if

one.

Whitnlf

Two curves

and only it they

have the same degree. Using calculus, the desr.e 18 detined as follows:

ex is an immersion, !la'il

+0

coordinate on sl) • Let

~. a t III II I

vector to a.

where da

II

= aldO

Ce

Since is the

be the unit tangent

Then 1 J211' DCa) • ~ ~d8.

o

We now give a combinatorial interpretation of' the Whitney degree, and relate this interpretation to the states of a universe. Definition 6.2.

When

C is an oriented Jordan curve in the

plane, define the curliness ~(C) - =1 &ecording as the curve is oriented clockwise (+1) or counterclockwise (-1). If U is any uniVerse, let

U be

the collection o~ Seifert

circl.es for U (see Lemma 3.4 and Figure l}). obtained from b

.

Thus U is

by splitting all vertices in oriented fash-

ion (compare Figure 30).

Define the curliness of U by the

formula: ~ (U)

= 8' (U) .. I:cEO ff (C).

98

Figure 30 Lemma.§.:2. U s a(Sl)

Let

a: sl

->

R2

be a normal immersion.

be the corresponding universe.

degree of a

Let

Then the Whitney

coincides with the curliness of U; that is,

D(a) = tr (U).

The (easy) proof of the lemma will be omitted.

Following

our combinatorial theme, it is of interest to give a proof that ~(U)

is invariant under regular homotopy.

As we see ~(U)

is

related to the topology of Jordan curves in the plane.

As

from the next lemma, this analysis will show how

usual, a configuration such as

denotes a universe

containing this local pattern, and a formula containing more than one such pattern refers to a single universe

th~

has

been changed locally to conform to the indicated patterns. Lemma 6.4.

---.;r

eo(

.rc--

~(

!.- ......" )

)

~(

J.\

- ft( 'L

) =

1

)

-1.

99 It suffices to Oheck these formulas for UftiYl•••• • t

Proof.

Jordan curves.

The lemma then follows from the char, ••~IW

and the chart we have not drawn. obtained trom the vlei_1, chart by reversing all the arrows.

J.t

~ ~

8 @ ~ Proposition~.

~(U)

e( ~t)

((~)

g

:1. -:1. -:2.

::2..

@ 00

0

-:1.

If U is regularly homotopic to U', then

= ff(u').

Proof.

=ff(~4)

==ff(;14) =

~(:x=?)

==

~(~)

-1

(lr ) +1-1 = f1 (;>C) .

.. eo

:. ff Thus

~

(~)

(6.4)

is invariant under moves of type A.

ff

(A)

== ff~~) =~\~~) = r,

The final case for the type next page.

\-)t-) B

move is illustrated on the

100

This completes the proof of Proposition 6.5.

Any universe is regularly homotopic to a disjoint union of 'the standard torms indicated in Figure 31.

For the reader

interested in constructing a combinatorial proof of this fact, the ''Whitney triCk". as illustrated in Figure· 32, will ,

be usef'Ul.. /

101

CJJOc)~ -a

-1..

1

0

standard

.

'"

~

Figure 31

Yo--~~~~.

~1l~1l Whitney Trick Figure 32 In order to relate the curliness of a universe to its states, it is convenient to use the string form.

Upon de-

composing a string (see section 2) into Seifert Circles, there will be a single string without self-crossings, and a collection of Jordan curves.

Assign curliness zero to the string

without self-crossings, and the usual plus or minus one to each Jordan curve.

Thus

102

a te s trom. an y st th e cu rU n es t ac tr ex to how We now show . o f a u n iv er se o f th e el is a la b el b la g in ss o a l cr .6 . A v e rt ic D ef in it io n 6 se U . V (U ) F or & un iv er . 1 .. AD where p la ci n g form ~ ob ta in ed by U f o g n la b el li th e v er ti ca l w il l de no te o ss in g . el a t ea ch cr b la l ca ti er e a v and V (U ) th n g un iv er se ri st a be hen L et U te o t U. T 'l'I!eorem §.:.1. be an y st a 8 t le ; U el li n g o t v er ti ca l la b

D is cu ss io n. ch U fo r whi

of is t st at es S ex e er th at rv e th We fi rs t ob se example: i8 tr u e. F or .7 6 : ot a ul th e fo rm

X

~

A l _ A~ (U) 2-

103

In order to construct a state for whi:ch first form

U,

th'

'l1l\I&1 ..... ,

the set of Seifert circles for U.

a trail by the following procedure: with empty interior.

Locate

.... ......

.i,.i.

a a.1t."

Reassemble one of its sites

resulting configuration has one fewer circle.

10

'ho

Repeat

process, never using a given site more than once.

th.

th~.

The

re.~'

is a trail whose corresponding state has the required prope,',. (The proof of this fact will be omitted.)

For example:

Trail with state markers.

Seifert circles with indicated re-assemblies.

The State S. Proof of 6.7.

By using the relation AD = 1

see that when S and

S'

it is easy to

are related by a sequence of state

transpositiO~S, then .. y ~z:

Crossing Codes

r~=i) Y!' = j" ZZ

-1-

~»

Ll. Let

a link L.

44

t>

Xl'~,

Assume that

••• ,~

~~

=1

be the code variables for for each index k.

(Thus

these codes are the modified Alexander codes of Lemma 6.8.)

[x{- ~~ ... X:nJ

Let

[JS.

~

~J

.)o.~

denote the difference al an n:.:al _an =Xl ... ~ + (-l).lI.l ... ~

As in the single variable case, we have inner products

and state polynomia1s



for a given link L

whose strands have been labelled with these code variables. However, it is now necessary to give a norma1ization factor since the unadorned state polynom1a1 is no longer independent

106 ac co mp lis h th is sta rs . In or de r to ot the ch oic e ot tix ed the un iex fo r the reg ion s ot ind ple lti mu a e us en d, we sh all ve cto r ind ex of U is as sig ne d a ve rse U. Each reg ion se s ,or de ) The kth ind ex , Pk , inc rea 'p = (P l'P 2" •• ,P n' ind ica ted in Fig ure the kt h str an d as cre as es upon cro ssi ng (0 ,0, ••• ,0 ). ion is as sig ne d ind ex 34 . The unbounded reg

p•

(~,...,1\.•...

x" Po)

r

p' • ( ... ... . I\. +'-•.••• p.l

M ult ipl e Ind ex ing Fig ure 34 sind ex ed un ive rse co rre Le t U be a mu lti ply cti on of sta tes to r Le t $ be th e co lle L. k lin a to ing nd po ion s wi th ind ice s are in a pa ir of reg rs sta ed fix ose wh U and pI = p = (P l,P 2, ••• ,Pn )

. Proposition~

kth pla ce . Le t sta rs . ot the ch oic e of fix ed /I$1 is ind ep en de nt

on ly in the Then

itt ed . It tol low s op os iti on wi ll be om The pr oo f ot th is pr pr oo f of Theorem 3. 3. the same' tor ma t as the we must tak e olO gic al inv ari an ce , In or de r to ob tai n top ven L as above, 6) in to ac co un t. Gi on cti se e (se ss ne cu rli lab ell ed su ,U Ln ' th e se pa rat ely ••• U L3 U ~ U Ll = le t L ell in g di sti nc t ve the op tio n of lab ha we at th ote (N . lin ks e va ria bl e.) Then str an ds wi th the sam

107

is a topological invariant of potential function of the Example

liD!

L.

We refer to

&.

U.

([5], [17]).

x

t =

1

(by convention)

It I • X(X-~), ~(L) • X- l • X :. DL = l/(X-X). Example

1.d.. p'S 0

181

= (X2 (-1»X(X-X)

DL

= (~(L)/ltl> = x/x2x- 1 (x-X)

• DL = l/(X-X).

DL u tne

loa

,_,. uK~ aD K.. 1K

l=i®,$)

= -If +

It/ _

(f2 'Oy2'(-1»i(y_i) = Y(Y-i)

OK = =

XY = x(y-i)

(X~(Kl\'t(~) lit/ ) (xy/y(y-y»x(y-i)

:. ,OK = 1.

Example L.§..

Thus we ha.ve

O( 0 ) = 1/(X-X) D( b) = 1/(Y-f)

D(ct)) = 1 and

by

a similar calcula.tion,

D(~)

= -1.

In general. it L is a single variable link (with variable

X assigned to all strands), then

109

where the Conway polynomial is regarded as a function ot z = X-X.

In ([5]) Conway uses the notation for the poten-

tial function that we have reserved for the Conway polynomial With this caveat, our normalizations coincide with his. The potential function still satisfies the basic identity

X

(X]

K

OK - OK

=

=X- X

[XJO L

for links related at crossings involving identically labelled strands. In addition to the basic identity there are also identities for crossings of differently labelled strands: Theorem Ll.

[XY] .. XY + Xi.

Let

Then

1)

D(~)

+

O(:>C. =:1:1 where

D L , = >.[Yl DL = :try] DL Y is the variable labelling the strand of L that as shown below.

Then

is linked by the new component.

rI

X Y

Proof.

Apply the definitions to the states and labels indi-

cated below.

\

The details of the calculation are omitted.

112

Example

1.:2.

str an d wi th

W ad din g a new be ob ta. ine d fro m L be low . number n, as shown

Ln

~t

J,.~k1ns

y

x.

t t (LO

Th en

OL = 0

o

OL

1

Dr.-xl Th us

(7 .a)

= [Y JO L

(7.7)

n '-'Ln _2

n _l = [x y]'-'L n

D~ = (xy ] 4 1

is sp lit )

...

[XY ](Y )

Dr.

-l) [Y JD r. O L - «(x y)2••• :3

Example 7.1 0.

-=>

L

O L = [xYJ - [X]{YJ.

De mo ns tra tio n.

w' -O L + O f: = (X ]O Sim il.a .rl y, O f: =

:. O L

= O f:

Ow = [YJ

(by 7.a )

[xY];

- [XHYJ

= {xYJ - [X ][Y ].

"

/

1l.3

Example 7.11.

B

B

DB

is known a.s the Ballantine rings.

ot its potential function.

= [X] [y}[Z] •

We omit the calcuJ.a.tion

(There are sixteen states in the

Ballantine Universe.) Example 7.12.

B'

Let this tamily inherit color and orientation trom the Ballantine rings in

~le

c:J B

=

7.11.

-[XYZ]

(calculation omitted).

DB' = 0

(split).

DB + OBI'" DB + OBI (Theorem 7.7) •

.. D B .=

-([XYZ]

+ [XJ[Y][Z}).

11 4

§.

Sp an nin g

S~tt&ces

al .ph ere . nte d) thr ee -di me ns ion rie (o the ote d~n Le t S3 is sa id to be a F embedded in s3 An or ien ted sll rfa ce K is id en tic al a kn ot or lin k K if sp an nin g su rfa ce fo r n on F ind uc es F, and the or ien tat io wi th the bo un da ry of n on K. the giv en or ien tat io r co ns tru cti ng a sp ec ifi c method fo In [36 J Se ife rt gave m. His th a giv en kn ot dia gra wi ted cia so as ce rfa a sp an nin g su rfa ce of the dia gra m. na ted the Se ife rt su su rfa ce wi ll be de sig s alg ori thm , re la te all de sc rib e Se ife rt' In th is se cti on we sh genus ot the Se inway po lyn om ial to the the de gre e of the Co tio n of the se re lacu ss the ge ne ral iza fe rt su rfa ce , and dis s vi a the Se ife rt ry sp an nin g su rfa ce tio ns hip s to ar bi tra pa iri ng . Se ife rt' s Al go rit hm

) on the kn ot di acle s (se e Fi gu re 13 Form th e Se ife rt cir as ind ica ted be low . arc s at ea ch cro ssi ng o tw ng wi dra by m gra

::K

ve rsi ng the di athe n ob tai ne d by tra are s cle cir rt ife Se The s when ar riv in g moving alo ng th e arc and s ge ed its ng alo gram (in the pla ne ) ch ci rc le 1s embedded ea us Th . ng ssi cro at a rt su rfa ce is cro ssi ng s. The Se ife the of nt me ple com in the Se ife rt cir cle s ac hin g dis cs to the att ) by d cte tru ns co the n ee thr the in tly in are embedded di sjo so th at the se dis cs ten din g above ot the dia gra m and. ex ne pla the ing lud inc sp ac e in a tw ist ed band co mp let ed by til lin g it. The su rfa ce is

110

single, dual, or, tri-colored (that is, there may be one, two, or three variables involved in the weave). Proof.

Since the potential function is a normalized state

summation, these identities can be demonstrated by the same method as used in Theorem 4.1.

In Figure 35 we have shown all

of the local state configurations with their contributions to the terms of state summations for the first identity (1). Four configurations cancel in pairs, leaving the pattern of this identity. ion.

The second identity follows in the same fash-

The verification of the third identity is a long calcu-

lation involving the triangle states (Figure 23) and is

+. ,.....-

- - T.........

L..

L\

~.&._.&._

YX

Xy

x:x.

YX

XY

~

YX

x9

::>'~

Xy

~

YX

Xy

YX

Xy

:::><x:.. " " :x::>c

Figure 35

~ Co

~ c

'\ /

L

116 are Note that the Seifer t circle s constr ucted in this proces s the same as those produc ed by splitti ng sites to preserv e orient ation. ThUS, in the example above, the c1.re.les drawn by site-s plittin g appear as

u

ClD

Seifer t Circle s

ted Each twisted band has· the same homotopy type as its projec image in the plane, a tilled -in crossin g:

1

pro' ••'

~ ::::I::

Conseq uently, the Seifer t surfac e has the homotopy type of the 2-cell complex obtain ed, from the univer se U underlying the knot diagram , by adding disjoi nt 2-ce11 s to the Seifer t circle s.

(Regard these cells as autom aticall y tilling

in the crossin gs.)

~

complex

117

Lemma 8.1.

Let K be a knot or link (diagram) with underlyinl

universe U.

Let

K, and let X{F) p{F)

F

be the Seifert surface}itor the diagram

denote the Euler characterl$tic of F, and

denote the rank of the first homology group

Hl{F).

Then X(F)

= s(u) -

R(U) + 2,

P(F) = R(U) - s(u) - 1, where R(U) = the number of regions in U. and number of Seifert circles in U.

s(u)

= the

Note that the Euler char-

acteristic and rank are functions of the universe alone. Proof. and

E

V = V(U)

Let 2

R = R(U)

EeU)

and

denote the number of vertices of U,

denote the number of edges ot U, while Then V - E + R = 2

S = S(U).

since the

plane has Euler characteristic 2, while the Euler characteristic ot the Seifert surface is given by

~(F) =

V- E+S

since the surface has the homotopy type of the two-cell plex constructed on the universe.

ooa-

(Recall that the Euler

characteristic of a two-complex is equal to NO - Nl + N. where

Ni

is the number of

i-cells in the

com~lex.

Rlaall

also that for a connected surface with boundary, the Eullr characteristic equals

2 - P where

first homologylgroup.)

p is the rank ot the

The lemma follows at once from thiS'

formulas. Remark.

A surface with boundary is said to have genus

g

if

it becomes a sphere with

g handles upon adding disks to all

the boundary components.

This gives rise to the relation

118

p = 2g + ponents.

- 1

~

tor a connected surface with we have the

Henc.~

Corollary 8.2.

Let

connected universe

K be a link diagram with U, R regions, and

Then the Seifert surface the fonnula ~.

boundary com-

~

g = i (R - S -

F ~)

for

1.1

oomponents,

S Seifert circles.

K has genus

g

liven by

•

It is of interest to note that a surface of relatively

low genus can arise from a diagram with many regions if these regions are balanced off by a goodly circles.

of Seifert

The simplest case is an unknot of the fonn

R

10

S

9

1.1 ~.

collec~ion

=

g

= i (10 - 9 - 1)

o.

1

Lest we give the impression that the Seifert surface

is easy to draw in perspective, please note that nested Seifert circles give rise to nested diSKS.

It is a well-known

bit of folklore that any knot has a diagram in its ambient isotopy class without nested Seifert circles.

I leave it to

the reader to make this concept of nesting preCise, but point out that of the two equivalent diagrams shown below tor the figure eight knot, only the second has a Seifert surface that can be drawn without overlapping disks (allowing one diSK to planar region) .

cv

119

cOPway

The Degree of the


S'-K

ing to the representation

(The Wl-

See Figure 49.)

has the following interpreta-

be the covering space correspondf: G -> Z.

The space X is the

infinite cyclic cover of the knot complement (see [26]). first homology group, Hl(X;Z), is a

r-module via the action

of the group of covering translations of X. ture, Hl(XjZ)

and

H(G.f)

The

are isomorphic

With this strucr-modules.

Th1s

interpretation is very important. but will not be pursued here. Returning to algebra, we wish to describe a presentation for

GI. and thence compute

presentation of the form: with

GI IG".

Suppose that

G has a

G = (s'gl'g2' ...• 9nfRl' •.. ,Rm)

160 1.

n .. "mi" ('1'ru. tor the Dehn presentation since there are' two

mo•••,,1ons

than crossings, and ,one region

corre.,ond. to the identity element.) 2.

or· m ~ n.

o.

f(e).~. t(ll) • f(g2) = ••• = f(~) =

The seoond aondttton 1s accomplished from an arbitrary presentation bJ ohooltns an the other ••n••,tol"

s

(f(s) = 1), and re-defining

via multiplication by approrpriate

power. ot ., to tnlure that they all hit zero under f. Iloall that r. Z[x,x- l ] acts on G via xg = SgS-l. It is eal, to .e. that

G'

is generated by the set

CxkS1/k • Z, i . l, .•• ,n}.

In particular, each relation Rk

can be rewritten in terms of these generators.

Let

p(Rk)

denote th"'s rewr1tlns of Rk • Then (~P(Rk)/k E Z, i - l, .•• ,m) is a Bet of relations tor G' (proof via covering spaces or combinatorial group theory). G'

= «(xkgi}/(xkp(Rj»)).

By

Thus

abelianizing these generators

and relations, and writing them additively, we obtain the structure of H(G,f). Consider the Dehn presentation. region of index 1

(or

-1

Let

s

if necessary).

correspond to a Suppose A, B,

C, D are the regions around a crossing with in
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2.

R. Ball and M.L. Mehta. Sequence of invariants for knots and links. (To appear in Journal de Physique.)

3.

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4.

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5.

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6.

J.H. Conway. Spring 1978.

7.

Daryl Cooper. Ph.D. Thes~s. Mathematics Institute, University of Warwick (1981).

8.

R.H. Crowell. Genus of alternating link types. Math. §2 (1959), 258-275.

9.

R.H. Crowell. Nonalternat1ng links. (1959), 101-120.

10.

R.H. Crowell and R.H. Fox. Introduction to Knot Theory. Blaisdell Publishing Company (1963). -- - -

11.

L. Euler. Solutio problematis ad geometriam situs pertinentis. Comment. Academiae Sci. I. Petro2ol1tanae 8 (1736},'128-l40. Opel' Omnia Series 1-7 (1766), 1-10.-

12.

R.H. Fox. A qu~ck trip through knot theory. Topology ot Three Manifolds. prentic~ Hall (1962), 120-167. --

13.

R.H. Fox and J.W. Milnor. Singularities of 2-spheres in 4-space and cobordism of knots. Osaka J. Math. 1 (1966), 257-267. \

14.

C.F. Gauss. Zur geometr1e del' lage fur zwe1 raum dimensionen, Werke, Band 8, 272-286, Konigl. Gesellshaft del' Wissenshaften, Goetigen, 1900.

15.

C. Giller. A family of links and the Conway calculus. (To appear.)

Lecture, University of Illinois at Chicago.

Ann. of

Ill. J. Math. 1

166 16.

I. Handler.

17.

R. Hartley. The Conway potential fUnction tor links. (To appear.)

18.

L.H. Kautf'man. 33-44.

19. 20.

(Private communication.)

Link manifolds.

Mich. Math. J. l i (1974),

L.H. Kauffman. Planar surface immersions. (1979). 648-665.

~

L.H. Kauff'man. Weaving patterns and polynomials. Topology Symposium Proceedings. Siegen 1979. Springer Verlag Lecture Notes in Mathematics ~, 88-97.

21. 'L.H. Kauffman., The Conway polynomial. (1980), 101-108. 22.

Ill. J. Math.

Topology;, 20

L.H. Kauff'man and T.F. Banchoft.' Immersions and mod-2 quadratic forms. Amer. Math. Monthly 84 (1977), 168-185. L.H. Kauffman and L. Taylor. Signature of links. Amer. Math. Soc. 216 (1976), 351-365.

Trans.

24.

J. Levine. Knot cobordism groups in codimension two. Comm. Math. Helv. 44 (1969). 229-24,4.

25.

J. Levine. Polynomial invariants of' knots of codimension two. Ann. of Math. 84 (1966), 534-554.

26.

J.W. Milnor. Infinite cyclic coverings. Topology of' Manifolds. (Mich. State Unlv. 1967). Prindle Weber-and Schmidt, Boston (1968).

27.

J.W. Milnor. Singular POints of Complex Hypersurfaces. Princeton University Press (1908).

28.

K. Murasugi. On the genus of the alternating knot I. J. Math. Soc. Japan !Q (1958), 94-105.

29.

K. Murasugi. On the genus of the alternating knot II. J. Math. Soc. Japan 10 (1958), 235-248.

30.

K. Murasugi. nating knot.

31.

K. Murasug1. A certain numerical invariant of link type. Trans. Amer. Math. Soc. 117 (1965), 387-422.

32.

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the Alexander polynomial of the alterOsaka J. Math. 10 (1958), 181-189. On

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H. Whitney. On regular closed curves in the pIe:ne. Math. i (1937), 276-284.

~

e:nd Links.

Comm.

Publish or Perish Press

~

Math. Ann.

Combinatorial Group

Department of Mathematics, Statistics e:nd Computer Science The University or Illinois at Chicago Chicago, Illinois 60680

Compo