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

David De Wit·Atsushi Ishii·Jon Links
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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.

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David De Wit, Atsushi Ishii, Jon Links. 2005-05-26. Infinitely many two-variable generalisations of the Alexander-Conway polynomial. https://doi.org/10.2140/agt.2005.5.405

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