Volume 5, issue 1 (2005)

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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 Δ is obtainable via a particular one-variable reduction of each two-variable Links–Gould invariant LGm,1, where m is a positive integer. Thus there exist infinitely many two-variable generalisations of Δ. This result is not obvious since in the reduction, the representation of the braid group generator used to define LGm,1 does not satisfy a second-order characteristic identity unless m = 1. To demonstrate that the one-variable reduction of LGm,1 satisfies the defining skein relation of Δ, 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
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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