The $\operatorname{SU}(N)$ Casson–Lin invariants for links

Hans U. Boden and Eric Harper

Vol. 285 (2016), No. 2, 257–282

DOI: 10.2140/pjm.2016.285.257

Abstract

We introduce the $SU\left(N\right)$ 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 $\left(a_1,\dots ,a_n\right)\in {\mathbb{Z}}^n$ of labels, where $a_j$ is associated with ${\ell }_j$. The $SU\left(N\right)$ Casson–Lin invariant, denoted $h_{N,a}\left(L\right)$, gives an algebraic count of certain projective $SU\left(N\right)$ representations of the link group ${\pi }_1\left(S^3\backslash L\right)$, and the family $h_{N,a}$ of link invariants gives a natural extension of the $SU\left(2\right)$ 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 $\left(a_1,\dots ,a_n\right)$, the invariants $h_{N,a}\left(L\right)$ vanish whenever $L$ is a split link.

Keywords

braids, links, representation spaces, Casson–Lin invariant

Mathematical Subject Classification 2010

Primary: 20C15, 57M25

Milestones

Received: 11 June 2015

Revised: 5 March 2016

Accepted: 18 May 2016

Published: 21 November 2016