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

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.