Applied Mathematics and Computation 153 (2004) 199–204

www.elsevier.com/locate/amc

A computer program to calculateAlexander polynomial from

Braids presentation of the given knot

Hakan S�ims�ek a,*, Mustafa Bayram a, U�gur Yavuz b

a Atat€urk €Universitesi, Fen Edebiyat Fak€ultesi, Matematik B€ol€um€u, 25240 Erzurum, Turkeyb Atat€urk €Universitesi, _I letisim Fak€ultesi, Gazetecilik B€ol€um€u, 25240 Erzurum, Turkey

Abstract

A central quantity for the calculation of Alexander polynomial of knots is to use

Braids presentations of the given knots. For this purpose, we improved a computer

program which is writting in Delphi programming language. The program calculates

Alexander polynomials of the given knot using free derivative that is obtained from

Braids presentation of the given knot.

� 2003 Elsevier Inc. All rights reserved.

Keywords: Alexander polynomial; Alexander matrix; Knot; Free derivative

1. Introduction

The Alexander Polynomial was discovered by James Waddel Alexander II in

1928 as a generalization of linear color tests. Designed to be calculated through

matrices, the original form differs from more recent polynomial invariants that

use skein relations [1–5]. John Conway, however, found a skein relation that

lead to The Alexander Polynomial in the 1960�s, bridging the gap between

Alexander�s polynomial and today�s widely used polynomial invariants [6–10].Skein relation very important role in polynomial invariants of Knot Theory.

Definition. A knot is an embedding of a circle S1 into Euclidean 3-space, R3 or

the 3-sphere, S3.

* Corresponding author.

E-mail addresses: hsimsek@atauni.edu.tr (H. S�ims�ek), bayrams@atauni.edu.tr (M. Bayram).

0096-3003/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/S0096-3003(03)00623-4

200 H. S�ims�ek et al. / Appl. Math. Comput. 153 (2004) 199–204

Definition (Tame knots). A knot K is called tame if it is ambient isotopic to a

simple closed polygon in R3 respectively in S3. A knot is wild if it is not tame.

Theorem 1 (J. Alexander). Any tame oriented link in 3-sace may be representedby a pair ðb; nÞ, where b is an element of the n-string braid group Bn. The link Lobtained by closing b, i.e., trying the top end of each string to the same position onthe bottom of the braid as shown in Fig. 1. The closed braid will be denoted by b.

Definition (Regular projection). A projection p of a knot K is called regular if

i(i) there are only finitely many multiple points fPij16 i6 ng, and all multiple

points are double points, that is, p�1ðPiÞ, contains two points;

(ii) no vertex of K is mapped onto a double point. (The minimal number of

double points or crossings n in a regular projection of a knot is called

the order of the knot.)

Theorem 2 (Braids presentation). Let ri, i ¼ 1; 2; . . . ; n be the over crossingarcs of a regular projection of a knot K. Then the knot group admits the followingso-called Braids representation:

G ¼ p1ðS3 � V ðKÞÞ ¼ hs1; s2; . . . ; snjr1; r2; . . . ; rni:

The arc ri corresponds to the generator si; a crossing of characteristic gi as inFig. 1 gives rise to the defining relator rj ¼ sjs

�gji s�1

k sgji .

An Example: K is a Trefoil knot. From Fig. 2 we obtain Braids generatorsand relaters s1s2s�1

3 s�12 at vertex A, s2s3s�1

1 s�13 at the vertex B, s3s1s�1

2 s�11 at the

vertex C. Since by Theorem 2, one relation is a consequence of other two knot

group has presentation.

hs1; s2; s3js1s2s�13 s�1

2 ; s3s1s�12 s�1

1 i ¼ hs1; s2js1s2s1s�12 s�1

1 s�12 i ¼ hx; yjx3y2i;

where y ¼ s�12 s�1

1 s�12 and x ¼ s1s2.

Fig. 1. Skein diagram.

Fig. 2. Regular diagram of Right-Hand-Trefoil.

H. S�ims�ek et al. / Appl. Math. Comput. 153 (2004) 199–204 201

We will construct some especially simple derivatives on group ring.

ZF ðfx1; x2; . . . ; xj; . . . ; xxgÞ of the free group F ðfx1; x2; . . . ; xj; . . . ; xxgÞ, whichwe will also denote by ZF and F , respectively.

Definition (Free derivatives). LetQk

s¼1xesis be an arbitrary (not necessarily re-

duced) word in the free group F , where every es ¼ �. We defines the free de-

rivatives by

o

oxj

Yks¼1

xesis ¼Xir

ð�1Þð1=2Þðer�1ÞYks<r

xesis xð1=2Þðer�1Þis

for all j.

2. System and methods

The program was written in Delphi. Using this program, it is possible to

calculate the Alexander polynomial of given knots from Braids presentation.

Definition (Alexander matrix). Let hx1; x2; . . . ; xnjr1; r2; . . . ; rmi be a finite pre-

sentation of a group G. Then the Alexander matrix associated with this pre-

sentation is the matrix� �� �

a h

orioxj

;

where h denotes the natural ring homomorphism.

ZF ðfx1; x2; . . . ; xj; . . . ; xxgÞ ! Zhx1; x2; . . . ; xnjr1; r2; . . . ; rmi

and a denotes the natural ring homomorphism

Zhx1; x2; . . . xnjr1; r2; . . . ; rmi !Zhx1; x2; . . . ; xnjr1; r2; . . . ; rmi=hx1; x2; . . . ; xnjr1; r2; . . . ; rmi:

202 H. S�ims�ek et al. / Appl. Math. Comput. 153 (2004) 199–204

Clearly this matrix depends on the particular group presentation which has

been chosen for the group G.Let A be an Alexander matrix which has entries from a commutative ring R

with unit element 1. If A is a m� n matrix then elementary ideal EðAÞ in R to be

ideal generated in R by all determinants of every ðn� 1Þ � ðn� 1Þ submatrix of

A. Suppose that the link associated with an n-braid r is a knot KðrÞ. Then we

have that GðKðrÞ � hx1; x2; . . . ; xnjx1rx�11 ; . . . ; xn�1rx�1

n�1i. Hence the Alexander

matrix of this group is the ðn� 1Þ � n matrix A ¼ aij� �

, where

aij ¼oðxjrx�1

j Þoxj

����������x1¼���¼xn¼t

for 16 i6 n� 1 and fundamental formulas,

Xnj¼1

aijðt � 1Þ ¼ ðxirx�1i Þ�1

x1¼���¼xn¼t ¼ 0 2 Z½t; t�1�:

Since xi�r ¼ AixilA�1i for all i and Z½t; t�1� has no divisors of zero, we have thatPn

j¼1 aij ¼ 0. So the Alexander matrix of GðKðrÞÞ is equivalent to the matrix

oðxj�rx�1j Þ

oxj

!ðx1¼���¼xn¼tÞ16 i;j6 ni;j6 n�1

on;1

24

35:

This shows that the elementary ideal is of the form f ðtÞ, where f ðtÞ is a poly-

nomial in t with integer coefficients and non-zero positive constant term when

f ðtÞ 6¼ 0. Such a polynomial is called the Alexander polynomial of the knotKðrÞ.

3. Application of the computer program

We applied the computer program to the Trefoil, Granny and Figure-eight

knots, and their braid presentations as follows:

Braid presentation of the right hand Trefoil knot:B3 ¼ hx1; x2jx1x2x1 ¼ x2x1x2i,

Braid presentation of the left hand Trefoil knot:

B3 ¼ hx1; x2jx2x1x2 ¼ x1x2x1i,Braid presentation of the Granny knot:

B4 ¼ hx1; x2; x3jx1x2x1 ¼ x2x1x2; x2x3x2 ¼ x3x2x3i andBraid presentation of the Figure-eight knot:

B4 ¼ hx1; x2; x3jx1 ¼ x1x2x�11 x3x1x�1

2 x�11 ; x2 ¼ x�1

3 x1x3i, respectively.

The computer program calculates free derivatives of the given knots using

Braid presentation of them and then calculates Alexander polynomials. The

computer results as follows.

Free derivatives DðtÞ(Alexander polynomials)

Right hand

Trefoil knot

oox1

ðx�11 x�1

2 x�11 x2x1x2Þ ¼ �x�1

1 � x�11 x�1

2 x�11 þ x�1

1 x�12 x�1

1 x2 DðtÞ ¼ 1� t þ t2

oox2

ðx�11 x�1

2 x�11 x2x1x2Þ ¼ �x�1

1 x�12 þ x�1

1 x�12 x�1

1 þ x�11 x�1

2 x�11 x2x1

Left hand

Trefoil knot

oox1

ðx�11 x�1

2 x�11 x2x1x2Þ ¼ �x�1

1 � x�11 x�1

2 x�11 þ x�1

1 x�12 x�1

1 x2 DðtÞ ¼ 1� t þ t2

oox2

ðx�11 x�1

2 x�11 x2x1x2Þ ¼ �x�1

1 x�12 þ x�1

1 x�12 x�1

1 þ x�11 x�1

2 x�11 x2x1

Granny knot oox1

ðx�11 x�1

2 x�11 x2x1x2Þ ¼ �x�1

1 � x�11 x�1

2 x�11 þ x�1

1 x�12 x�1

1 x2 DðtÞ ¼ ð1� t þ t2Þ2oox2

ðx�11 x�1

2 x�11 x2x1x2Þ ¼ �x�1

1 x�12 þ x�1

1 x�12 þ x�1

1 þ x�11 x�1

2 x�11 x2x1

oox2

ðx2x3x2x�13 x�1

2 x�13 Þ ¼ x2x3 � x2x3x2x�1

3 x�12

oox3

ðx2x3x2x�13 x�1

2 x�13 Þ ¼ x2 � x2x3x2x�1

3 � x2x3x2x�13 x�1

2 x�13

Figure-eight

knot

oox1

¼ ðx�12 x1x2x�1

1 x2x1x�12 x�1

1 x2x�11 Þ ¼ x�1

2 � x�12 x1x2x�1

1 þ x�12 x1x2x�1

1 x2�x�1

2 x1x2x�11 x2x1x�1

2 � x�12 x1x2x�1

1 x2x1x�12 x�1

1 x2x�11

DðtÞ ¼ 1� 3t þ t2

H.S�im

s� eket

al./Appl.Math.Comput.153(2004)199–204

203

204 H. S�ims�ek et al. / Appl. Math. Comput. 153 (2004) 199–204

References

[1] J.W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928)

275–306.

[2] K. Murasugi, On the braid index of alternating link, Trans. Amer. Math. Soc. 123 (1991) 237–

260.

[3] H.R. Morton, Seifert circles and knot polynomials, Math. Proc. Cambridge Phil. Soc. 99

(1986) 107–109.

[4] H.R. Morton, P.R. Cromwell, Distinguishing mutants by knot polynomials, J. Knot Theory

Ramif. (1996) 225–238.

[5] H.R. Morton, The multivariable Alexander polynomials for closed braids, Contemporary

Math. 233, Amer. Math. Soc., 1999, pp. 167–172.

[6] J.S. Birman, D. Long, J. Moody, Linear representation of the braid group, Contemporary

Math. 169 (1994) 123–132.

[7] S. Moran, The Mathematical Theory of Knots and Braids, North Holland Math. Studies, vol.

82, 1983.

[8] M. Bayram, Alexander Polynomials, M.Sc. thesis, Atat€urk University, Erzurum, 1988.

[9] H. S�ims�ek, Positive Knots have Positive Conway Polynomials, Ph.D. thesis, Atat€urk

University, Erzurum, 2001.

[10] M. Bayram, H. S�ims�ek, N. Yıldırım, Automatic calculation of Alexander polynomials of

ð3; kÞ-torus knots, Appl. Math. Comp. 136 (2003) 505–510.