Harer-Zagier formulas for families of twisted hyperbolic knots

TL;DR

This paper derives Harer-Zagier formulas for certain families of twisted hyperbolic knots, revealing factorized structures and intriguing zero patterns, advancing the understanding of knot invariants and matrix models.

Contribution

It introduces new formulas for the Harer-Zagier transform of HOMFLY-PT polynomials for infinite twisted hyperbolic knot families, including a fully factorized Pretzel knot family.

Findings

Factorized form of the Harer-Zagier transform for some Pretzel knots

Zeros of the transform have modulus of their product equal to one

Transform consists of sums of factorized terms for other families

Abstract

In an attempt to generalise knot matrix models for non-torus knots, which currently remains an open problem, we derived formulas for the Harer-Zagier transform of the HOMFLY-PT polynomial for some infinite families of twisted hyperbolic knots. Among them, we found a family of Pretzel knots for which the transform has a fully factorised form, while for the remaining families considered it consists of sums of factorised terms. Their zeros have a remarkable structure as the modulus of their product always equals unity.