arXiv subjects
Publications and source records attributed to P. Traczyk.
The problem of faithfulness of the (reduced) Burau representation for $n =4$ is known to be equivalent to the problem of whether certain two matrices $A$ and $B$ generate a free group of rank two. It is known that $A^3$ and $B^3$ generate a free group of rank two \cite{9}, \cite{10}, \cite{4}. We prove that they also generate a free group when considered as matrices over the $\mathbb{Z}_p[t,t^{-1}]$ for any integer $p > 1$.
\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.