A diagrammatic computation of abelian link invariants

Cimasoni, David; Ferretti, Livio; Liu, Jessica

doi:10.2140/agt.2025.25.5113

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A diagrammatic computation of abelian link invariants

David Cimasoni, Livio Ferretti and Jessica Liu

Algebraic & Geometric Topology 25 (2025) 5113–5136

DOI: 10.2140/agt.2025.25.5113

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 (2021), which was recently proven to compute the Levine–Tristram signature and the Alexander polynomial of oriented links (see Liu, 2023 and Cimasoni and Ferretti, 2024). As a corollary, we obtain a multivariable extension of Kauffman’s (1983) determinantal model of the Alexander polynomial, recovering a result of Zibrowius (2017).

Keywords

link diagrams, multivariable signature, multivariable Alexander polynomial

Mathematical Subject Classification

Primary: 57K10

References

Publication

Received: 27 May 2024

Revised: 4 October 2024

Accepted: 30 December 2024

Published: 20 November 2025

Authors

David Cimasoni
Section de mathématiques Université de Genève Genève Switzerland

Livio Ferretti
Section de mathématiques Université de Genève Genève Switzerland

Jessica Liu
Department of Mathematics University of Toronto Toronto, ON Canada

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Volume 25, issue 8 (2025)