the burau representation of the braid group was introduced in the 1930s and has been used since the 1980s to understand the jones polynomial. a critical component of this study has been trying to understand when the burau representation is faithful. it is known to be faithful for the braid group on fewer than 4 strands and unfaithful on more than 4 strands. if the representation is unfaithful on 4 strands, it will solve the jones unknot conjecture. we will review the current state of knowledge and some new approaches suggested by the work of armond, huynh and l\^e.