Conway's potential function via the Gassner representation

TL;DR

This paper presents a new method to compute Conway's multivariable potential function using the Gassner representation, generalizing previous work and resolving sign ambiguities in related formulas.

Contribution

It introduces a formula connecting the potential function with the reduced Gassner representation, improving computational efficiency and clarifying existing definitions.

Findings

Provides an explicit formula for the potential function via Gassner representation

Removes sign ambiguity in formulas relating Alexander polynomial and Gassner representation

Relates different definitions of the reduced Gassner representation in literature

Abstract

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.