# On the sheaf-theoretic SL(2,C) Casson-Lin invariant

## TL;DR

This paper proves that the sheaf-theoretic SL(2,C) Casson-Lin invariant is generally independent of a parameter and additive under connected sums of knots, using topology, microlocal analysis, and algebraic geometry.

## Contribution

It establishes the parameter independence and additivity of the sheaf-theoretic SL(2,C) Casson-Lin invariant, addressing key open questions.

## Key findings

- Invariant is generically independent of the parameter τ.
- Invariant is additive under connected sums of knots.
- Uses Behrend functions to interpret the invariant.

## Abstract

We prove that the ($\tau$-weighted, sheaf-theoretic) SL(2,C) Casson-Lin invariant introduced by Manolescu and the first author in [CM19] is generically independent of the parameter $\tau$ and additive under connected sums of knots in integral homology 3-spheres. This addresses two questions asked in [CM19]. Our arguments involve a mix of topology, microlocal analysis and algebraic geometry, and rely crucially on the fact that the SL(2,C) Casson-Lin invariant admits an alternative interpretation via the theory of Behrend functions.