Knots from the random matrix theory with a replica

Abstract

A classical knot is described by a one-stroke trajectory with entanglements of a string. The replica method appears as a powerful tool in statistical mechanics for a polymer or self-avoiding walk. We consider this replica N to 0 limit in Gaussian means of the products of trace of N x N Hermitian matrices, which provides one-stroke graphs of a knot. The Seifert surfaces of knots and links are derived by a random matrix model. The zeros of Alexander polynomials on a unit circle are discussed for the case of n-vertices in the analogy of Yang-Lee edge singularity. The extension of one matrix model is considered for higher dimensional knots and for half integral level k in Chern-Simons gauge theory.