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How the Links-Gould invariants generalise the Alexander-Conway polynomial
by
David De Wit
The University of Queensland
Coauthors: Atsushi Ishii and Jon Links

For any positive integer m, the Alexander-Conway polynomial D is obtainable as a particular one-variable `root-of-unity' reduction of the two-variable Links-Gould invariant LG^{m, 1}. This follows as the reduction of LG^{m, 1} satisfies the defining skein relation of D. This is nontrivial as the reduced representation of the relevant braid generator X does not itself satisfy this relation. Instead, the key to the elegant little proof of this result involves determining the kernel of a quantum trace. We are able to evaluate this kernel from knowledge of the representation underlying LG^{m, 1} without having to determine X explicitly.

For positive integers m, n, strong circumstantial evidence supports a conjectured nonlinear version of this result: LG^{m, n} reduces to a power of D. This conjecture is all the more interesting as we currently have no method of explicitly evaluating even LG^{2, 2} for any links other than the closures of 2-braids.

Date received: January 19, 2005

Copyright © 2005 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # capo-07.