Perverse sheaves and knot contact homology

In this paper, we give a new algebraic construction of knot contact homology in the sense of Ng [35]. For a link L in R-3, we define a differential graded (DG) k-category (A) over tilde (L) with finitely many objects, whose quasi-equivalence class is a topological invariant of L. In the case when Lis a knot, the endomorphism algebra of a distinguished object of (A) over tilde (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 B-n 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) over tilde (L) = H-0((A) over tilde (L)) defined as the 0-th homology of (A) over tilde (L) is equivalent to the category of perverse sheaves on R-3 that are singular along the link L. We also obtain several generalizations of the category (A) over tilde (L) by extending the Gelfand-MacPherson-Vilonen braid group action. Detailed proofs of results announced in this paper will appear in [4]. (C) 2017 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.