A sheaf-theoretic SL(2,C) Floer homology for knots

Abstract

Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three-manifolds, similar to the three-manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat connections with fixed holonomy around the meridian of the knot. Thus, our invariant is a sheaf-theoretic SL(2,C) analogue of the singular knot instanton homology of Kronheimer and Mrowka. We prove that for two-bridge and torus knots, the SL(2,C) invariant is determined by the l-degree of the $\widehat{A}$-polynomial. However, this is not true in general, as can be shown by considering connected sums of knots.