Ihara zeta function and twisted Alexander invariants

TL;DR

This paper extends the interpretation of knot invariants as Ihara zeta functions by providing a zeta function expression for twisted Alexander invariants, building on previous work relating knot polynomials to cycle counting.

Contribution

It introduces an analogous zeta function formulation for twisted Alexander invariants, expanding the conceptual framework linking knot invariants and cycle counting.

Findings

Provides a zeta function expression for twisted Alexander invariants

Builds on previous models relating knot polynomials to Ihara zeta functions

Enhances understanding of knot invariants through cycle counting methods

Abstract

Lin and Wang defined a model of random walks on knot diagrams and interprete the Alexnader polynomials and the colored Jones polynomials as Ihara zeta functions, i.e. zeta functions defined by counting cycles on the knot diagram. Using this explanation, they gave a more conceptual proof for the Melvin-Morton conjecture. In this paper, we give an analogous zeta function expression for the twisted Alexander invariants.