Strands algebras and Ozsváth and Szabó's Kauffman-states functor

Abstract

We define new differential graded algebras $\mathcal{𝒜}\left(n,k,\mathcal{𝒮}\right)$ in the framework of Lipshitz, Ozsváth and Thurston’s and Zarev’s strands algebras from bordered Floer homology. The algebras $\mathcal{𝒜}\left(n,k,\mathcal{𝒮}\right)$ are meant to be strands models for Ozsváth and Szabó’s algebras $\mathcal{ℬ}\left(n,k,\mathcal{𝒮}\right)$; indeed, we exhibit a quasi-isomorphism from $\mathcal{ℬ}\left(n,k,\mathcal{𝒮}\right)$ to $\mathcal{𝒜}\left(n,k,\mathcal{𝒮}\right)$. We also show how Ozsváth and Szabó’s gradings on $\mathcal{ℬ}\left(n,k,\mathcal{𝒮}\right)$ arise naturally from the general framework of group-valued gradings on strands algebras.