Weights of Markov traces for Alexander polynomials of mixed links

TL;DR

This paper introduces a new approach to defining Alexander polynomials for mixed links using Fourier expansion of Markov traces, revealing a simple form of coefficients and relating mixed link invariants to resolved links.

Contribution

It provides a direct definition of Alexander polynomials for mixed links via Fourier expansion of Markov traces, connecting these invariants to simpler resolved links.

Findings

Fourier coefficients of Markov traces are simple under specialization

Alexander polynomial of mixed links relates to that of resolved links

New algebraic framework for mixed link invariants

Abstract

Using the Fourier expansion of Markov traces for Ariki-Koike algebras over $\mathbb{Q}(q,u_{1},...,u_{e})$, we give a direct definition of the Alexander polynomials for mixed links. We observe that under the corresponding specialization of a Markov parameter, the Fourier coefficients of Markov traces take quite simple form. As a consequence, we show that the Alexander polynomial of a mixed link is essentially equal to the Alexander polynomial of the link obtained by resolving the twisted parts.