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

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

2004-05-21

arxiv.org/abs/math/0405403v6

Algebr. Geom. Topol. 5 (2005) 405-418

Mathematics

Geometric Topology

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-18.abs.html

Scientific paper

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.

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