Perverse sheaves and knot contact homology

Abstract

In this paper, we give a new algebraic construction of knot contact homology in the sense of Ng [35]. For a link L in R3, we define a differential graded (DG) k-category A˜L with finitely many objects, whose quasi-equivalence class is a topological invariant of L. In the case when L is a knot, the endomorphism algebra of a distinguished object of A˜L coincides with the fully noncommutative knot DGA as defined by Ekholm, Etnyre, Ng, and Sullivan in [13]. The input of our construction is a natural action of the braid group Bn on the category of perverse sheaves on a two-dimensional disk with singularities at n marked points, studied by Gelfand, MacPherson, and Vilonen in [19]. As an application, we show that the category of finite-dimensional representations of the link k-category A˜L=H0(A˜L) defined as the 0-th homology of A˜L is equivalent to the category of perverse sheaves on R3 that are singular along the link L. We also obtain several generalizations of the category A˜L by extending the Gelfand–MacPherson–Vilonen braid group action. Detailed proofs of results announced in this paper will appear in [4].