Khovanov-Seidel quiver algebras and Ozsv\'ath-Szab\'o's bordered theory

TL;DR

This paper establishes a connection between Khovanov-Seidel quiver algebras and Ozsváth-Szabó's bordered theory, showing isomorphisms and homotopy equivalences that relate their algebraic structures in knot theory.

Contribution

It demonstrates that a quotient of Ozsváth-Szabó's algebra is isomorphic to Khovanov-Seidel's quiver algebra and relates their bimodules via induction and restriction, linking two prominent theories.

Findings

A quotient of Ozsváth-Szabó's algebra is isomorphic to Khovanov-Seidel's quiver algebra.

Khovanov-Seidel's bimodule is homotopy equivalent to Ozsváth-Szabó's DA bimodule after scalar induction and restriction.

The results connect two algebraic frameworks used in knot theory and bordered Floer homology.

Abstract

We investigate a relationship between Ozsv\'ath and Szab\'o's bordered theory and the algebras and bimodules constructed by Khovanov-Seidel. Specifically, we show that (a variant of) a special case of Ozsv\'ath-Szab\'o's algebras has a quotient which is isomorphic to the Khovanov-Seidel quiver algebra with coefficients in $\mathbb{Z}/2\mathbb{Z}$. Furthermore, we show that after induction and restriction of scalars, the dg bimodule over quiver algebras associated to a crossing by Khovanov-Seidel is homotopy equivalent to Ozsv\'ath-Szab\'o's DA bimodule for the crossing in this special case.