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A mnemonic for the Lipshitz–Ozsváth–Thurston correspondence

Artem Kotelskiy, Liam Watson and Claudius Zibrowius

Algebraic & Geometric Topology 23 (2023) 2519–2543

When k is a field, type D structures over the algebra k[u,v](uv) are equivalent to immersed curves decorated with local systems in the twice-punctured disk. Consequently, knot Floer homology, as a type D structure over k[u,v](uv), can be viewed as a set of immersed curves. With this observation as a starting point, given a knot K in S3, we realize the immersed curve invariant HF^(S3 \ ν (K)) of Hanselman, Rasmussen and Watson by converting the twice-punctured disk to a once-punctured torus via a handle attachment. This recovers a result of Lipshitz, Ozsváth and Thurston calculating the bordered invariant of S3 \ ν (K) in terms of the knot Floer homology of K.

Fukaya categories of punctured surfaces, bordered Heegaard Floer theory, multicurve invariants, knot Floer homology
Mathematical Subject Classification
Primary: 57K18, 57K31
Secondary: 57R58
Received: 30 July 2020
Revised: 8 October 2021
Accepted: 11 February 2022
Published: 7 September 2023
Artem Kotelskiy
Department of Mathematics
Indiana University
Bloomington, IN
United States
Department of Mathematics
Stony Brook University
Stony Brook, NY
United States
Liam Watson
Department of Mathematics
University of British Columbia
Vancouver, BC
Claudius Zibrowius
Faculty of Mathematics
University of Regensburg
Department of Mathematical Sciences
Durham University
United Kingdom

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