Knots from the random matrix theory with a replica

TL;DR

This paper explores the application of the replica method from statistical mechanics to random matrix models for describing classical knots, their Seifert surfaces, and properties of Alexander polynomials, extending to higher dimensions and gauge theories.

Contribution

It introduces a novel connection between replica limits in random matrix theory and knot theory, including the derivation of Seifert surfaces and analysis of Alexander polynomial zeros.

Findings

Replica N to 0 limit yields knot graph structures.

Zeros of Alexander polynomials are on the unit circle.

Extension to higher-dimensional knots and Chern-Simons theory.

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.