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Satellite knots and
immersed Heegaard Floer homology
Wenzhao Chen and Jonathan Hanselman |
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Geometry & Topology 30 (2026) 1719–1828
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Abstract |
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We describe a new method for computing the knot Floer complex of a satellite knot given the knot Floer complex for the companion and a doubly pointed bordered Heegaard diagram for the pattern, showing that the complex for the satellite can be computed from an immersed doubly pointed Heegaard diagram obtained from the Heegaard diagram for the pattern by overlaying the immersed curve representing the complex for the companion. This method streamlines the usual bordered Floer method of tensoring with a bimodule associated to the pattern by giving an immersed curve interpretation of that pairing, and computing the module from the immersed diagram is often easier than computing the relevant bordered bimodule. In particular, for (1,1)-patterns the resulting immersed diagram is genus one, and thus the computation is combinatorial. For (1,1)-patterns this generalizes previous work of Chen which showed that such immersed Heegaard diagrams compute the knot Floer complex of the satellite. As a key technical step, which is of independent interest, we extend the construction of a bigraded complex from a doubly pointed Heegaard diagram and of an extended type-D structure from a torus-boundary bordered Heegaard diagram to allow Heegaard diagrams containing an immersed alpha curve. |
Keywords
satellite knots, Heegaard Floer homology, immersed curves
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Mathematical Subject Classification
Primary: 57K18
Secondary: 57R58
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References |
Publication
Received: 12 February 2024
Revised: 25 February 2026
Accepted: 26 March 2026
Published: 18 July 2026
Proposed: Ciprian Manolescu
Seconded: András I Stipsicz, Peter Ozsváth
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Authors |
| Wenzhao Chen | |
| Institute of Mathematical
Sciences ShanghaiTech University Shanghai China |
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| Jonathan Hanselman | |
| Department of Mathematics Indiana University Bloomington Bloomington, IN United States |
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| © 2026 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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