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

TL;DR

This paper investigates the faithfulness of the reduced Burau representation for n=4, demonstrating that matrices A^2 and B^2 generate a free group of rank 2, advancing understanding of this open problem.

Contribution

It shows that A^2 and B^2 generate a free group of rank 2, providing a new approach towards proving the faithfulness of the Burau representation for n=4.

Findings

A^2 and B^2 generate a free group of rank 2

Progress towards the faithfulness problem for n=4

Refinement of previous results by Breidze and Traczyk

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.