Perverse Sheaves and Knot Contact Homology

In this paper, which is mostly a research announcement, we give a new algebraic construction of knot contact homology in the sense of L. Ng [Ng05a]. For a link $L$ in ${\mathbb R}^3$, we define a differential graded (DG) $k$-category $\tilde{\mathscr A}$ 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 $\tilde{\mathscr A}$ coincides with the fully noncommutative knot DGA as defined by Ekholm, Etnyre, Ng and Sullivan in [EENS13a]. 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 [GMV96]. As an application, we show that the category of finite-dimensional representations of the link $k$-category $\tilde{A} = H_0(\tilde{\mathscr A})$ defined as the $0$th homology of our DG category $\tilde{\mathscr A}$ is equivalent to the category of perverse sheaves on ${\mathbb R}^3$ which are singular along the link $L$. We also obtain several generalizations of the category $\tilde{\mathscr A}$ by extending the Gelfand-MacPherson-Vilonen braid action.

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