On the Burau representation of the braid group $B_4$

Abstract

In this thesis, we establish strong constraints on the kernel of the (reduced) Burau representation $\beta_4:B_4\to \text{GL}_3\left(\mathbb{Z}\left[q^{\pm 1}\right]\right)$ of the braid group $B_4$, addressing a conjecture originally posed in the 1930s. The strategy of the proof is a concrete interpretation of $\beta_4\left(\sigma\right)$ in terms of the Garside normal form for $\sigma\in B_4$. More specifically, if $\sigma$ is a positive braid in $B_4$ satisfying certain constraints, then we show that $\beta_4\left(\sigma\right)$ is not a diagonal matrix by considering a new decomposition of positive braids and combinatorially interpreting $\beta_4\left(\sigma\right)$ in terms of this decomposition.