  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.
