Honors Thesis: On the faithfulness of the Burau representation at roots of unity

Abstract

We study the kernel of the evaluated Burau representation through the braid element $\sigma_i \sigma_{i+1} \sigma_i$. The element is significant as a part of the standard braid relation. We establish the form of this element's image raised to the $n^{th}$ power. Interestingly, the cyclotomic polynomials arise and can be used to define the expression. The main result of this paper is that the Burau representation of the braid group of $n$ strands for $n \geq 3$ is unfaithful at any primitive root of unity, excepting the first three.