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.