The Image of the Gassner Representation of the Pure Braid Subgroup has Pairwise Free Generators

TL;DR

This paper explores the Gassner representation of the pure braid group, establishing its relation to the Colored-Burau representation and simplifying analysis through linear algebra techniques.

Contribution

It demonstrates the equivalence of the Colored-Burau and Gassner representations on the pure braid subgroup, providing a new approach for their analysis.

Findings

Colored-Burau representation is equivalent to Gassner on pure braids

Analysis reduces to basic linear algebra techniques

Provides new insights into the structure of pure braid representations

Abstract

While much is known about the faithfulness of the Burau representation, the problem remains open for the Gassner representation for every $B_n$ with $n\geq 4$. We first find the definition of the Colored-Burau representation of Ainshel, Ainshel, Goldfeld, and Lemieux and we show that this is equivalent, when restricted to the pure braid subgroup, to the Gassner representation. The methods of Abdulrahim and Knudson require analysis within the lower central series of a free subgroup of the pure braid group. However, Lipschutz's work gives a method for analyzing the Gassner representation and the Colored-Burau structure reduces this analysis to basic linear algebra.