SL(2,C) Floer Homology for Knots and Knot Surgeries

We investigate the sheaf-theoretic SL(2,C) Floer cohomology for knots and 3-manifolds presented as surgeries on knots in S^3. We establish a relationship between the 3-manifold invariants, HP(Y) and HP#(Y) for Y a surgery on a small knot in S^3, and the SL(2,C) Casson invariant defined by Curtis. We use this to compute HP for surgeries on the trefoil and the figure-eight knots. We also compute HP for surgeries on two non-small knots, the granny and square knots. For the knot invariant, we prove that the (τ-weighted, sheaf-theoretic) SL(2,C) Casson-Lin invariant is generically independent of the parameter τ and additive under connected sums of knots in integral homology 3-spheres. This addresses two questions posed by Cote-Manolescu