Strands algebras and Ozsv\'ath-Szab\'o's Kauffman-states functor

TL;DR

This paper introduces new differential graded algebras as strands models for Ozsváth-Szabó's algebras in bordered Floer homology, establishing a quasi-isomorphism and elucidating grading structures.

Contribution

It defines new strands algebras A(n,k,S), proves their quasi-isomorphism to Ozsváth-Szabó's B(n,k,S), and clarifies the origin of gradings within this framework.

Findings

Established a quasi-isomorphism between A(n,k,S) and B(n,k,S)

Demonstrated how Ozsváth-Szabó's gradings arise from group-valued gradings

Extended the strands algebra framework within bordered Floer homology

Abstract

We define new differential graded algebras A(n,k,S) in the framework of Lipshitz-Ozsv\'ath-Thurston's and Zarev's strands algebras from bordered Floer homology. The algebras A(n,k,S) are meant to be strands models for Ozsv\'ath-Szab\'o's algebras B(n,k,S); indeed, we exhibit a quasi-isomorphism from B(n,k,S) to A(n,k,S). We also show how Ozsv\'ath-Szab\'o's gradings on B(n,k,S) arise naturally from the general framework of group-valued gradings on strands algebras.