Harer-Zagier formulas for knot matrix models

TL;DR

This paper extends Harer-Zagier formulas to knot matrix models, revealing factorization properties for torus knots and discussing implications for eigenvalue model constructions of knot invariants.

Contribution

It generalizes Harer-Zagier formulas to knot matrix models and analyzes their behavior for different knots, highlighting special properties of torus knots.

Findings

Harer-Zagier formulas for torus knots factorize nicely.

Factorization does not occur for non-torus knots.

Connection between factorization and eigenvalue model construction for knots.

Abstract

Knot matrix models are defined so that the averages of characters are equal to knot polynomials. From this definition one can extract single trace averages and generation functions for them in the group rank - which generalize the celebrated Harer-Zagier formulas for Hermitian matrix model. We describe the outcome of this program for HOMFLY-PT polynomials of various knots. In particular, we claim that the Harer-Zagier formulas for torus knots factorize nicely, but this does not happen for other knots. This fact is mysteriously parallel to existence of explicit beta = 1 eigenvalue model construction for torus knots only, and can be responsible for problems with construction of a similar model for other knots.