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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
Abstract

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.

Keywords
Fukaya categories of punctured surfaces, bordered Heegaard Floer theory, multicurve invariants, knot Floer homology
Mathematical Subject Classification
Primary: 57K18, 57K31
Secondary: 57R58
References
Publication
Received: 30 July 2020
Revised: 8 October 2021
Accepted: 11 February 2022
Published: 7 September 2023
Authors
Artem Kotelskiy
Department of Mathematics
Indiana University
Bloomington, IN
United States
Department of Mathematics
Stony Brook University
Stony Brook, NY
United States
https://artofkot.github.io/
Liam Watson
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada
https://personal.math.ubc.ca/~liam/
Claudius Zibrowius
Faculty of Mathematics
University of Regensburg
Regensburg
Germany
Department of Mathematical Sciences
Durham University
Durham
United Kingdom
https://cbz20.raspberryip.com/

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