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

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

QUANDLE TWISTED ALEXANDER INVARIANTS

We establish a quandle version of the twisted Alexander polynomial. We also develop a theory that reduces the size of a twisted Alexander matrix with column relations. The reduced matrix can be used to refine invariants derived from the twisted Alexander matrix