Khovanov–Seidel quiver algebras and Ozsváth–Szabó's bordered theory

Abstract

In the 2016 preprint “Kauffman states, bordered algebras, and a bigraded knot invariant,” Ozsváth and Szabó introduced a set of algebraic constructions in the spirit of bordered Heegaard Floer homology. Their constructions can be used to compute knot Floer homology algebraically for knots in the 3-sphere. In this paper we investigate a relationship between Ozsváth–Szabó's bordered theory and the algebras and bimodules constructed by Khovanov and Seidel in “Quivers, Floer cohomology, and braid group actions” (2002). Specifically, we show that the Khovanov–Seidel quiver algebras are isomorphic to quotients of idempotent truncations of some of Ozsváth–Szabó's algebras. Furthermore, we show that the dg bimodule associated to a braid generator by Khovanov–Seidel, with the right action restricted using the quotient map, is homotopy equivalent to Ozsváth–Szabó's DA bimodule with the left action induced using the quotient map.