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 A3
and B3 generate the free group of rank 2. In our work, we show that
A2 and B2 generate the free group of rank 2