#### Volume 5, issue 1 (2005)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Infinitely many two-variable generalisations of the Alexander–Conway polynomial

### David De Wit, Atsushi Ishii and Jon Links

Algebraic & Geometric Topology 5 (2005) 405–418
 arXiv: math.GT/0405403
##### Abstract

We show that the Alexander-Conway polynomial $\Delta$ is obtainable via a particular one-variable reduction of each two-variable Links–Gould invariant $L{G}^{m,1}$, where $m$ is a positive integer. Thus there exist infinitely many two-variable generalisations of $\Delta$. This result is not obvious since in the reduction, the representation of the braid group generator used to define $L{G}^{m,1}$ does not satisfy a second-order characteristic identity unless $m=1$. To demonstrate that the one-variable reduction of $L{G}^{m,1}$ satisfies the defining skein relation of $\Delta$, we evaluate the kernel of a quantum trace.

##### Keywords
link, knot, Alexander-Conway polynomial, quantum superalgebra, Links–Gould link invariant
##### Mathematical Subject Classification 2000
Primary: 57M25, 57M27
Secondary: 17B37, 17B81
##### Publication
Received: 21 January 2005
Revised: 14 April 2005
Accepted: 28 April 2005
Published: 22 May 2005
##### Authors
 David De Wit Department of Mathematics The University of Queensland 4072 Brisbane Australia Atsushi Ishii Jon Links Department of Mathematics The University of Queensland 4072 Brisbane Australia