  In this paper we investigate the construction of state models for link
invariants using representations of the braid group obtained from various gauge
choices for a solution of the trigonometric Yang-Baxter equation. Our results
show that it is possible to obtain invariants of regular isotopy (as defined by
Kauffman) which may not be ambient isotopic. We illustrate our results with
explicit computations using solutions of the trigonometric Yang-Baxter equation
associated with the one-parameter family of minimal typical representations of
the quantum superalgebra U_q[gl(2|1)]. We have implemented Mathematica code to
evaluate the invariants for all prime knots up to 10 crossings.
