  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.
