On the considerations adopted by Breidze and Traczyk towards the faithfulness of Burau representation for $n=4$

Abstract

This work discusses the open problem of the faithfulness of the reduced Burau representation for $n=4$. Birman showed that in order to prove this representation is faithful, it is sufficient to find two matrices $A$ and $B$ that generate a free group of rank 2. Breidze and Traczyk proved that $A^3$ and $B^3$ generate the free group of rank 2. In our work, we show that $A^2$ and $B^2$ generate the free group of rank 2.

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