  In order to obtain a Markov theorem without stabilization, Birman and Menasco
introduced the notion of exchange related braids. In this paper I study the way
the Fiedler polynomial distinguishes conjugacy classes of some particular
braided knots. I introduce the Kauffman bracket in the solid torus. Its Taylor
expansion give finite type invariants similar to the Fiedler polynomial. I
investigate how these invariants distinguish exchange related braids.
