Casson-Lin's invariant of a knot and Floer homology

Details Casson-Lin's invariant of a knot and Floer homology Casson-Lin's invariant of a knot and Floer homology

1996-05-22

arxiv.org/abs/q-alg/9605036v1

Mathematics

Quantum Algebra

22 pages, AmsLaTex

Scientific paper

A. Casson defined an intersection number invariant which can be roughly thought of as the number of conjugacy classes of irreducible representations of $\pi_1(Y)$ into $SU(2)$ counted with signs, where $Y$ is an oriented integral homology 3-sphere. X.S. Lin defined an similar invariant (signature of a knot) to a braid representative of a knot in $S^3$. In this paper, we give a natural generalization of the Casson-Lin's invariant to be (instead of using the instanton Floer homology) the symplectic Floer homology for the representation space (one singular point) of $\pi_1(S^3 \setminus K)$ into $SU(2)$ with trace-free along all meridians. The symplectic Floer homology of braids is a new invariant of knots and its Euler number of such a symplectic Floer homology is the negative of the Casson-Lin's invariant.

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