Casson-Lin's Invariant and Floer Homology

Abstract

Casson defined an invariant which can be thought of as the number of conjugacy classes of irreducible representations of π1(Y) into SU(2) counted with signs, where Y is an oriented integral homology 3-sphere. Lin defined a similar invariant (the signature of a knot) for a braid representative of a knot in S3. In this paper, we give a natural generalization of Casson-Lin's invariant. Our invariant is the symplectic Floer homology for the representation space of π1(S3 \ K) into SU(2) with trace-zero along all meridians. The symplectic Floer homology of braids is a new invariant of knots and its Euler number is the negative of Casson-Lin's invariant.