The Yang-Baxter Equation in Knot Theory

The role played by the Yang-Baxter equation in generating knot invariants using the method of statistical mechanics is reexamined and elucidated. The formulation of knot invariants is made precise with the introduction of piecewise-linear lattices and enhanced vertex and interaction-round-a-face (IRF) models with strictly local weights. It is shown that the Yang-Baxter equation and the need of using its solution in the infinite rapidity limit arise naturally in the realization of invariances under Reidemeister moves III. It is also shown that it is essential for the vertex models be charge-conserving, and that the construction of knot invariants from IRF models follows directly from the vertex model formulation.