Link Invariants Associated with Gauge Equivalent Solutions of the Yang-Baxter Equation: the One-Parameter Family of Minimal Typical Representations of U_q[gl(2|1)]

Abstract

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.