Laboratoire de Mathématiques Jean Leray

The Casson-Lin invariant h(K) was originally defined in 1992 for knots K in S^3 by X.-S. Lin as an algebraic count of irreducible SU(2) representations of the knot group with meridional trace equal to 0, and he proved h(K) = sign(K)/2, one half the signature of K. The invariant was extended to other trace conditions and to knots in homology 3-spheres by C. Herald in 1997 and independently by M. Heusener and J. Kroll in 1998. More recently, E. Harper and N. Saveliev use projective SU(2) representations to define a Casson-Lin type invariant for 2-component links L in S^3, which they identify as the linking number of the two components of L.

This talk is a report on recent joint work with Eric Harper defining a family of analogous invariants in terms of irreducible projective SU(N) representations of the link group for links L in S^3. We will explain the compactness and irreducibility results needed to show the invariants are well-defined, and we outline how to compute the invariants for Hopf links and chain links, and we will present a vanishing result for split links.