Knot Theory based on Solvable Models at Criticality

Abstract

A general theory is presented to construct representations of the braid group and link polynomials (topological invariants for knots and links) from exactly solvable models in statistical mechanics at criticality. Sufficient conditions for the existence of the Markov trace are explicitly shown. Application of the theory to IRF and vertex models yields various link polynomials including an infinite sequence of new invariants. The new link polynomials are extended into two-variable link invariants. For the models with crossing symmetry, braid-monoid algebras associated with the link polynomials are derived. It is found that the Yang-Baxter relation gives both an algebraic approach and a graphical approach in knot theory.