Casson-Lin Type Invariants for Links

• Main Author: Harper, Eric
• Format: Others
• Published: Scholarly Repository 2010
• Subjects: SU(2) Representations Instanton Floer Homology
• Online Access: http://scholarlyrepository.miami.edu/oa_dissertations/372

In 1992, Xiao-Song Lin constructed an invariant h of knots in the 3-sphere via a signed count of the conjugacy classes of irreducible SU(2)-representations of the fundamental group of the knot exterior with trace-free meridians. Lin showed that h equals one-half times the knot signature. Using methods similar to Lin's, we construct an invariant of two-component links in the 3-sphere. Our invariant is a signed count of conjugacy classes of projective SU(2)-representations of the fundamental group of the link exterior with a fixed 2-cocycle and corresponding non-trivial second Stiefel--Whitney class. We show that our invariant is, up to a sign, the linking number. We further construct, for a two-component link in an integral homology sphere, an instanton Floer homology whose Euler characteristic is, up to sign, the linking number between the components of the link. We relate this Floer homology to the Kronheimer-Mrowka instanton Floer homology of knots. We also show that, for two-component links in the 3-sphere, the Floer homology does not vanish unless the link is split.