Perverse Sheaves and Knot Contact Homology

TL;DR

This paper introduces a new algebraic approach to knot contact homology using perverse sheaves and DG categories, providing topological invariants and linking to representation categories of links.

Contribution

It constructs a DG category invariant of links via perverse sheaves and braid group actions, connecting knot contact homology with sheaf theory and representation categories.

Findings

The DG category $ ilde{ extbf{A}}$ is a topological invariant of links.

The category of finite-dimensional representations of $ ilde{ extbf{A}}$ matches perverse sheaves on $ extbf{R}^3$ singular along the link.

The construction generalizes previous knot contact homology frameworks.

Abstract

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.