#### Volume 20, issue 7 (2020)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Strands algebras and Ozsváth and Szabó's Kauffman-states functor

### Andrew Manion, Marco Marengon and Michael Willis

Algebraic & Geometric Topology 20 (2020) 3607–3706
##### Abstract

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

##### Keywords
strands algebras, Floer homology, bordered Floer homology, sutured manifolds, bordered sutured Floer homology, A-infinity algebras, Kauffman states, knot Floer homology, tangle Floer homology, bordered knot Floer homology
##### Mathematical Subject Classification 2010
Primary: 57M25, 57M27, 57R56
Secondary: 57R58