Harer-Zagier formulas for families of twisted hyperbolic knots

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.