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.

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David De Wit, Atsushi Ishii and Jon Links
DDW and JL: Department of Mathematics, The University of Queensland
4072, Brisbane, Australia
AI: Department of Mathematics, Graduate School of Science, Osaka University
Machikaneyama 1-16, Toyonaka, Osaka, 560-0043, Japan

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