Algebraic invariants of knots and disordered Potts model

Abstract

The authors propose the reformulation of the Kauffman bracket invariant of the knot in terms of statistical mechanics of the 2D disordered Potts model. This allows one to put the question of the determination of knot entropy (or of the probability of an arbitrary knot formation) in terms of usual statistical mechanics. To demonstrate the possibilities of their approach they give a constructive estimation for the trivial knot formation probability for a long strongly contracted closed random path confined in a thin slit. They use the mean-field approximation for the free energy of the Potts system at the point of the transition from the paramagnetic phase to the spin-glass one.