Untitled

Abstract

We introduce the SU(N) Casson-Lin invariants for links L in S 3 with more than one component. Writing L = 1 ∪ • • • ∪ n , we require as input an n-tuple (a 1 , . . . , a n ) ∈ ‫ޚ‬ n of labels, where a j is associated with 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 π 1 (S 3 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 , . . . , a n ), the invariants h N,a (L) vanish whenever L is a split link.