Knot Theory and Integrable Systems

Abstract

This review article concerns the construction of link polynomials, topological invariants for knots and links, from exactly solvable(integrable) models. Through a general theory which is based on fundamental properties of the models, representations of the braid group and the Markov traces on the representations are made. In addition, the equivalence of algebraic and graphical formulations is proved. Various examples including Alexander, Jones, Kauffman and new link polynomials are explicitly shown. To sum up, the soliton theory contains the essence of the knot theory.