On the sheaf-theoretic $\operatorname{SL}(2,\mathbb{C})$ Casson–Lin invariant

Abstract

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