2016/10/07 by Yuri Berest, Berest, Yuri, Alimjon Eshmatov +3
Mathematics · #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1610.02438
openalex publication_date 2016/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 R3 , we define a differential graded (DG) k-category \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 \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 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 [GMV96]. As an application, we show that the category of finite-dimensional representations of the link k-category A = H0(\mathscr A) defined as the 0th homology of our DG category \mathscr A is equivalent to the category of perverse sheaves on \mathbb R3 which are singular along the link L . We also obtain several generalizations of the category \mathscr A by extending the Gelfand-MacPherson-Vilonen braid action.