  The recent proof by Bigelow and Krammer that the braid groups are linear
opens the possibility of applications to the study of knots and links. It was
proved by the first author and Menasco that any closed braid representative of
the unknot can be systematically simplified to a round planar circle by a
sequence of exchange moves and reducing moves. In this paper we establish
connections between the faithfulness of the Krammer-Lawrence representation and
the problem of recognizing when the conjugacy class of a closed braid admits an
exchange move or a reducing move.
