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

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