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

Weiping Li
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Abstract

A. Casson defined an intersection number invariant which can be roughly thought of as the number of conjugacy classes of irreducible representations of $π_1(Y)$ 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 $π_1(S^3 \setminus K)$ 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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Weiping Li. 1996-05-22. Casson-Lin's invariant of a knot and Floer homology. https://arxiv.org/abs/q-alg/9605036

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