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

Andrew Manion, Marco Marengon and Michael Willis

Algebraic & Geometric Topology 20 (2020) 3607–3706

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

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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
Received: 7 September 2019
Revised: 8 March 2020
Accepted: 26 March 2020
Published: 29 December 2020
Andrew Manion
Department of Mathematics
University of Southern California
Los Angeles, CA
United States
Marco Marengon
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA
United States
Michael Willis
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA
United States