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.
Categories: math.GR math.RT