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

De Wit, David; Ishii, Atsushi; Links, Jon

doi:10.2140/agt.2005.5.405

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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

DOI: 10.2140/agt.2005.5.405

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 $L G^{m, 1}$ , 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 $L G^{m, 1}$ 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 $Δ$ , 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

Volume 5, issue 1 (2005)