Exactly solvable models and knot theory

Abstract

Presented is a review on theory of exactly solvable models in statistical mechanics and its application to knot theory. The Yang-Baxter relation, a sufficient condition for the solvability of models, is introduced for scattering matrices in (1 + 1)-dimensional field theory and for Boltzmann weights of vertex models and IRF models in two-dimensional statistical mechanics. A systematic study of solutions of the Yang-Baxter relation shows that there exists at least an infinite number of two-dimensional exactly solvable models in classical statistical mechanics. The result implies that each universality class has at least one exactly solvable model. A novel connection between physics and mathematics is exposed. Namely, a general theory to derive link polynomials, topological invariants for knots and links, from the exactly solvable models is presented. It is emphasized that the Yang-Baxter relation is a key to relate various new developments in recent theoretical physics.