On Solvable Lattice Models and Knot Invariants

Abstract

Recently, a class of solvable interaction round the face lattice models (IRF) were constructed for an arbitrary rational conformal field theory (RCFT) and an arbitrary field in it. The Boltzmann weights of the lattice models are related in the extreme ultra violet limit to the braiding matrices of the rational conformal field theory. In this note we use these new lattice models to construct a link invariant for any such pair of an RCFT and a field in it. Using the properties of RCFT and the IRF lattice models, we prove that the invariants so constructed always obey the Markov properties, and thus are true link invariants. Further, all the known link invariants, such as the Jones, HOMFLY and Kauffman polynomials arise in this way, along with giving a host of new invariants, and thus also a unified approach to link polynomials. It is speculated that all link invariants arise from some RCFT, and thus the problem of classifying link and knot invariants is equivalent to that of classifying two dimensional conformal field theory.