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 $L{G}^{m,1}$, 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 $L{G}^{m,1}$ 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 $\Delta$, we evaluate the kernel of a quantum trace.