Casson-lin’s Invariant of a Knot and Floer Homology
نویسنده
چکیده
A. Casson defined an intersection number invariant which can be roughly 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. X.S. Lin defined an similar invariant (signature of a knot) to a braid representative of a knot in S. 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 π1(S \ 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.
منابع مشابه
A Volume Form on the Su(2)-representation Space of Knot Groups
For a knot K in S, we construct according to Casson—or more precisely taking into account Lin’s [Lin92] and Heusener’s [Heu03] further works—a volume form on the SU(2)-representation space of the group of K (see Section 3). We next prove that this volume form is a topological knot invariant (see Section 4) and explore some of its properties (see Section 5). Motivation and Main ideas In 1985, A....
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