Infinitely many two-variable generalisations of the Alexander-Conway polynomial
David De Wit, Atsushi Ishii, Jon Links
Abstract
We show that the Alexander-Conway polynomial Delta is obtainable via a particular one-variable reduction of each two-variable Links-Gould invariant LGm,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 LGm,1 does not satisfy a second-order characteristic identity unless m=1. To demonstrate that the one-variable reduction of LGm,1 satisfies the defining skein relation of Delta, we evaluate the kernel of a quantum trace.
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