A multivariable Casson-Lin type invariant

TL;DR

This paper introduces a new multivariable invariant for links in three-dimensional space, connecting representation theory with link signatures and exploring deformations of SU(2) representations.

Contribution

It defines a novel multivariable Casson-Lin type invariant for links, linking it to multivariable signatures and analyzing representation deformations.

Findings

Invariant equals sum of multivariable signatures for certain links

Provides new insights into SU(2) representation deformations

Establishes a signed count of irreducible representations

Abstract

We introduce a multivariable Casson-Lin type invariant for links in $S^3$. This invariant is defined as a signed count of irreducible $\operatorname{SU}(2)$ representations of the link group with fixed meridional traces. For 2-component links with linking number one, the invariant is shown to be a sum of multivariable signatures. We also obtain some results concerning deformations of $\operatorname{SU}(2)$ representations of link groups.