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Home , Alexander polynomial, Braid group

Contemp orary

The Multivariable Alexander for a Closed

H R Morton

Abstract A simple multivariable version of the reduced Burau matrix is

constructed for any braid It is shown howthemultivariable Alexander p oly

nomial for the closure of the braid can b e found directly from this matrix

Intro duction

It has been known for some time that the Alexander p olynomial of a closed

braid can b e found from the Burau matrix This relation was extended in

to presentthevariable Alexander p olynomial t x of the consisting

A

of the closed braid together with its axis A as the characteristic p olynomial

B t of the reduced n n Burau matrix B t of the braid detI x

The Alexander p olynomial of can be recovered by applying the TorresFox

formula to the variable p olynomial to get the equation

t t

A

n

t t

In this pap er I give a similar metho d for nding the multivariable Alexander

p olynomial of a link L presented as the closure of a braid The main ingredi

ent is a readily constructed multivariable version of the reduced Burau matrices

Other versions of coloured Burau matrices have b een develop ed for example by

Penne whichcanbeinterpreted as determining linear presentations of suitably

extended versions of the braid

The most useful feature of the matrices which are used here is that they are

extremely simple to rememb er and they give an immediate and very straightforward

construction of the Alexander p olynomial of a closed braid and axis leading at

once to the p olynomial of the closed braid For a pure braid the resulting matrix is

conjugate to a reduced version of the Gassner matrix the construction given here

has the advantage that it applies to anybraid which presents the link and do es

not require the braid to b e rewritten in any sp ecial form

An implementation of this calculation by a Maple pro cedure which returns

the multivariable Alexander p olynomial of given the braid was made in early

Mathematics Subject Classication Primary M

This pap er was prepared during the lowdimensional top ology program at MSRI Berkeley

in I am grateful to MSRI for their supp ort

c

copyright holder

H R MORTON

t 1 t 4 t 3

t 2 t 1

t 4 t 2

t 1 t 2 t 3 t 4

  

Figure The lab elled braid

byaLiverp o ol MSc student Julian Ho dgson and is usually available from the

Liverp o ol theory websitenhttpnnwwwlivacuknPureMathsnknotshtml

The multivariable Alexander p olynomial

Given a homomorphism G H from the group G of an oriented link L

L L

to an ab elian group H wecanuseFoxs free dierential calculus on a presentation of

G to nd an invariantofL which lies in the group ring ZH The full multivariable

L



and is Alexander p olynomial arises when H is the ab elianisation G G

L L

L

the natural pro jection In this case H is the free ab elian group on k generators

t t say where L has k comp onents L L and x t for every

k k i i



is a Laurent oriented meridian x of the comp onent L Then ZG G

i i L L

L

p olynomial in t t Any other homomorphism G H factors through a

k L



H and the resulting invariantisgiven by substituting G G homomorphism

L

L

the images of t in

i L

A coloured Burau matrix for Lab el the individual strings of

B by t t putting the lab el t on the string which starts from the p oint j at

n n j

the b ottom Figure shows this lab elling for the braid

  

B

Write C a for the n n matrix which diers from the unit matrix

i

only in the three places shown on row ifor i n

B C

B C

B C

B C

B C

B C

a a

C a

i

B C

B C

B C

B C

A

When i ori n the matrix is truncated appropriately to givetwo nonzero

entries in row i

THE MULTIVARIABLE FOR A CLOSED BRAID

Now construct the colouredreducedBurau matrix B t t of the general

n

braid

l

Y

r

i

r

r

C a in which a is the lab el of the current undercrossing as a pro duct of matrices

i

string This gives

l

Y

r

a B t t C

r n i

r

r

where a is the lab el of the undercrossing string at crossing r counted from the top

r

of the braid

  

In the example shown where the lab els a a

are t t t t t t t resp ectively and B is the matrix pro duct

  

C t C t C t C t C t C t C t

Each braid determines a p ermutation S by the representation of B on

n n

S in which a string connects p osition j at the b ottom to p osition j atthetop

n

In the example ab ove

Theorem The multivariable Alexander polynomial where A is the

A

axis of the closed nbraid is given by the characteristic polynomial det I

B t t with the identications t t x

n j

j

Remarks Supp ose that a link L is presented as the closure of a braid

on n strings and that some homomorphism G H is given Lo ok at the part

L

of the diagram of L which consists of The oriented meridians x x for the

n

strings at the b ottom of determine elements x H Ateach p oint further

j

up the braid the meridian of a string which starts at the b ottom as string j will b e

mapp ed to the same elementinH Furthermore when the braid is closed to form

L the strings j and j are identied so that x x

j

j

When is the pro duct of k disjoint cycles then the link L has k comp onents

The variables in the resulting p olynomial are x and a k element subset of t t

n

after the identications have b een made In the case k the substitution t

t t in the matrix B gives the standard reduced Burau matrix B t for

n

The discussion of the Alexander p olynomials of a link with k comp onents

can b e done most uniformly in terms of the Alexander invariant D of the link

L

dened by

for k

L

t

D

L

L

for k

t

The TorresFox formula gives the Alexander invariant of a sublink L of a link L C

in terms of the invariantforL C It says that

t

LC

D t

L

c

where the meridian of the curve C to be suppressed is replaced by and c is

the element represented bythecurve C in the complementof L ab elianised appro

priately Thus we can calculate the Alexander invariantofL from theorem

H R MORTON

g 1 g 2

g n-1

Figure The generators in the punctured disk

by suppressing the axis and applying the TorresFox formula Put x andnote

that At t t in the complementofLtoget

n

B t t detI

n

D

L

t t t

n

making anyidentications t t required where strings of b elong to the same

i

i

comp onentof L In the case when L has one comp onent this gives the wellknown

B t det I

t quoted earlier formula t

L

n

t

Proof of Theorem Apply Foxs free dierential calculus to a presenta

tion of the of the closed braid and axis very much as in

or The heart of the pro of lies in relating the fundamental group of the n

punctured disk spanning the axis A which meets at the b ottom of to the cor

resp onding group for the disk at the top of Write x x for the generators

n

of the group at the bottom represented by meridian lo ops around the punctures

and X X for the meridian lo ops around the punctures at the top It is well

n

known that X X can be expressed in terms of x x as X F x

n n i i

where F F F is an automorphism of the F determined bythe

n n n

braid Furthermore the map F from the B to Aut F is itself

n n

a Then F is the comp osite of elementary automorphisms

corresp onding to the elementary making up whicharegiven explicitly by



F x x F x x x x and F x x j i i

i i i i i j j

i i i

i

The group of the link A is presented by a generator x arising from a meridian



of A and generators x x as ab ove with n relations F x x x x The

n i i

reduced Burau matrix shows up most naturally by using a dierent system of

generators g g for F dened recursively by g x g x g The

n n i i i

element g can be represented by a lo op in the disk which encircles the rst i

i

punctures as indicated in gure The automorphism F then satises F g

i

i i



g g j i x g x g g g g and F F

j j i i i i i i

i i

i

The standard metho d for nding the Alexander invariant of a link or knot L

evaluated in ZH using a homomorphism G H starting from a presentation

L

of G by n generators fg g and n relations fr g is the following Calculate

L j i

r

i

the n n matrix of free derivatives and evaluate the entries in ZH

g

j

by applying Then delete one column corresp onding to a generator csay with

THE MULTIVARIABLE ALEXANDER POLYNOMIAL FOR A CLOSED BRAID

c and divide the determinant of the remaining matrix by c The result

is the evaluation of the Alexander invariant that is the multivariable Alexander

p olynomial when L has more than one comp onent or t twhenL is a

L

r s

knot Where a relation is written in the form r s the entries work

g

j

equally well in the matrix

The presentation of G by generators g g and x ab ovewillgive an im

L n

F g

i

mediate calculation of the Alexander invariant once weknow the entries

g

j

The chain rule for free derivatives shows that if we write and set

G F g thenF g F G and

i i i i



n

X

F G G F g

k i i



g G g

j k j

k

F G F g

k i

as the pro duct of matrices We can then compute the matrix

g G

j j



where runs through the elementary braids in

i

Nowwhen wehave

i



F G G G G

i i i

i

F G G j i

j j

Then

F G

i



G G

i

i

G

i

F G

i



G G

i

i

G

i

F G

i

G

i

F G

k

otherwise

jk

G

j

Apply to the matrix to get

C B

C B

C B

C B

C B

F G

k

C B

a a

C B

G

j

C B

C B

C B

A

H R MORTON



where a G G This is the value of on the meridian at the undercrossing

i

i



in the current elementary braid Similarly when we get the matrix

i

i

C B

C B

C B

C B

C B

 

C B

a a

C B

C B

C B

C B

A



with a G G which is again the value of on the meridian of the under

i

i

crossing string

The generator g is clearly unchanged byany automorphism F so the last row

n

in each of the matrices of partial derivatives is The leading n n

submatrix for the partial derivatives of F g is then the pro duct of the leading

i

submatrices from the constituent elementary braids and so is the reduced colured

Burau matrix of intro duced earlier with t x Thus

j j

F g

B t t v

i

n

B t t say

n

g

j

for some column v The matrix of free derivatives arising from the presenta

tion of the group by generators g g and x given ab ove is the n n

n

matrix whose rst n columns corresp onding to the generators g g are

n

 

F g x g x These columns form the matrix B t t x I

i i n n

g

j

They arise from the full matrix of free derivatives by deleting the column corre

sp onding to the generator x so the Alexander invariantforthelinkisthedeter



minant of this matrix divided by x Now clearly detB t t x I

n n

n

x detx B t t I x Then detI xB t t making the

n n

identications of variables t t forced by matching the strings at the top and

j

j

b ottom is the Alexander p olynomial of the closed braid and its axis a p ower

of x

The Alexander p olynomial of the closed braid itself can then b e found from the

TorresFoxformula as shown ab ove

References

JSBirman Braids links and mappingclass groups Ann Math Studies Princeton

Univ Press

RHCrowell and RHFox Intro duction to Ginn and Co Boston

Springer Graduate Texts

HRMorton Exchangeable braids In Low Dimensional Top ology London Mathematical

So ciety Lecture Notes ed R A Fenn Cambridge University Press

RPenne Multivariable Burau matrices and lab eled line congurations JKnot Theory

Ramif

Department of Mathematical Sciences University of Liverpool Liverpool L

BX England

Email address mortonlivacuk

URL httpwwwlivacuksu