Forks, Noodles and the Burau representation for $n=4$

Abstract

\begin{abstract} The reduced Burau representation is a natural action of the braid group $B_n$ on the first homology group $H_1({\tilde{D}}_n;\mathbb{Z})$ of a suitable infinite cyclic covering space ${\tilde{D}}_n$ of the $n$--punctured disc $D_n$. It is known that the Burau representation is faithful for $n\le 3$ and that it is not faithful for $n\ge 5$. We use forks and noodles homological techniques and Bokut--Vesnin generators to analyze the problem for $n=4$. We present a Conjecture implying faithfulness and a Lemma explaining the implication. We give some arguments suggesting why we expect the Conjecture to be true. Also, we give some geometrically calculated examples and information about data gathered using a C\texttt{++} program.