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

While much is known about the faithfulness of the Burau representation, the problem remains open for the Gassner representation for every $B_n$ with $n\ge 4$. 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.