#### Algebraic and Geometric Topology 5 (2005),
paper no. 18, pages 405-418.

## Infinitely many two-variable generalisations of the Alexander-Conway polynomial

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

**Abstract**.
We show that the Alexander-Conway polynomial Delta is obtainable via
a particular one-variable reduction of each two-variable Links-Gould
invariant LG^{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 LG^{m,1} does not satisfy a
second-order characteristic identity unless m=1. To demonstrate that
the one-variable reduction of LG^{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 invariant

**AMS subject classification**.
Primary: 57M25, 57M27.
Secondary: 17B37, 17B81.

**DOI:** 10.2140/agt.2005.5.405

**E-print:** `arXiv:math.GT/0405403`

Submitted: 21 January 2005.
(Revised: 14 April 2005.)
Accepted: 28 April 2005.
Published: 22 May 2005.

Notes on file formats
David De Wit, Atsushi Ishii and Jon Links

DDW and JL: Department of Mathematics, The University of Queensland

4072, Brisbane, Australia

and

AI: Department of Mathematics, Graduate School of Science, Osaka University

Machikaneyama 1-16, Toyonaka, Osaka, 560-0043, Japan

Email: Dr_David_De_Wit@yahoo.com.au, aishii@cr.math.sci.osaka-u.ac.jp, jrl@maths.uq.edu.au

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