The SU(N) Casson-Lin invariants for links

TL;DR

This paper introduces the $SU(N)$ Casson-Lin invariants for links in $S^3$, extending previous invariants for knots and 2-component links, and computes their values for specific links, showing they vanish for split links under certain conditions.

Contribution

It defines a new family of $SU(N)$ Casson-Lin invariants for links, generalizing prior invariants and providing explicit computations and vanishing results.

Findings

Computed invariants for the Hopf link and chain links.

Established vanishing of invariants for split links under mild conditions.

Extended the $SU(2)$ Casson-Lin invariant to $SU(N)$ for links.

Abstract

We introduce the $SU(N)$ Casson-Lin invariants for links $L$ in $S^3$ with more than one component. Writing $L = \ell_1 \cup \cdots \cup \ell_n$, we require as input an $n$-tuple $(a_1,\ldots, a_n) \in {\mathbb Z}^n$ of labels, where $a_j$ is associated with $\ell_j$. The $SU(N)$ Casson-Lin invariant, denoted $h_{N,a}(L)$, gives an algebraic count of certain projective $SU(N)$ representations of the link group $\pi_1(S^3 \smallsetminus L)$, and the family $h_{N,a}$ of link invariants gives a natural extension of the $SU(2)$ Casson-Lin invariant, which was defined for knots by X.-S. Lin and for 2-component links by Harper and Saveliev. We compute the invariants for the Hopf link and more generally for chain links, and we show that, under mild conditions on the labels $(a_1, \ldots, a_n)$, the invariants $h_{N,a}(L)$ vanish whenever $L$ is a split link.