On the Burau representation for $n=4$

A. Beridze, P. Traczyk

Published 2019-04-26Version 1

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. In [Ber-Tra] we gave a simple proof that $(A^3, B^3)$ is a free group of rank two, the result known earlier from [Wil-Zar]. In this paper we use a combination of methods of linear algebra and homology theory (the forks and noodles approach) to give another proof that $(A^3, B^3)$ is a free group.