Conway’s potential function via the Gassner representation

We show how Conway’s multivariable potential function can be constructed using braids and the reduced Gassner representation. The resulting formula is a multivariable generalization of a construction, due to Kassel–Turaev, of the Alexander–Conway polynomial in terms of the Burau representation. Apart from providing an efficient method of computing the potential function, our result also removes the sign ambiguity in the current formulas which relate the multivariable Alexander polynomial to the reduced Gassner representation. We also relate the distinct definitions of this representation which have appeared in the literature.

Keywords

knot, link, braid, Alexander–Conway polynomial, potential function, Burau representation, Gassner representation

2010 Mathematics Subject Classification

20F36, 57M25, 57M27

Full Text (PDF format)

Received 18 December 2017

Accepted 26 April 2019

Published 21 August 2020