On the Burau representation for $n=4$

A. Beridze, P. Traczyk

Published 2017-05-07Version 1

The reduced Burau representation is a natural action of the braid group $B_n$ on the first homology group $H_1({\tilde{D}}_n;Z)$ of a suitable infinite cyclic covering space ${\tilde{D}}_n$ of the $n$-punctured disk $D_n$ (see [2], [3]). It is known that Burau representation is faithful for $n\le 3$ (see [1],[2]) and it is not faithful for $n\ge 5$ (see [2], [6], [7]). In this paper, we use Bokut--Vesnin generators [4] and techniques developed in [3], 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-program.