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

TL;DR

This paper investigates the faithfulness of the Burau representation at roots of unity, showing it is unfaithful for most cases when the number of strands is at least three, with exceptions.

Contribution

It provides a detailed analysis of the kernel of the evaluated Burau representation at roots of unity, revealing unfaithfulness for most cases and connecting cyclotomic polynomials to the representation's properties.

Findings

Burau representation is unfaithful at roots of unity for n ≥ 3

Cyclotomic polynomials are used to analyze the representation

Unfaithfulness exceptions are only for the first three strands

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.