A diagrammatic computation of abelian link invariants

TL;DR

This paper introduces a diagrammatic method to compute multivariable link invariants like the signature and Alexander polynomial from a symmetric matrix derived from a link diagram, extending previous models and results.

Contribution

It provides a unified diagrammatic approach to compute multivariable link invariants using a single symmetric matrix, generalizing existing models and linking to recent proofs.

Findings

Multivariable signature and Alexander polynomial can be computed from a symmetric matrix.

The method generalizes Kashaev's single-variable matrix to multivariable cases.

A multivariable extension of Kauffman's Alexander polynomial model is obtained.

Abstract

We show how the multivariable signature and Alexander polynomial of a colored link can be computed from a single symmetric matrix naturally defined from a colored link diagram. In the case of a single variable, it coincides with the matrix introduced by Kashaev in [arXiv:1801.04632], which was recently proven to compute the Levine-Tristram signature and the Alexander polynomial of oriented links [arXiv:2311.01923, arXiv:2310.16729]. As a corollary, we obtain a multivariable extension of Kauffman's determinantal model of the Alexander polynomial, recovering a result of Zibrowius [arXiv:1601.04915v1].