#### Volume 5, issue 1 (2005)

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

### David De Wit, Atsushi Ishii and Jon Links

Algebraic & Geometric Topology 5 (2005) 405–418
 arXiv: math.GT/0405403
##### 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.