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The cosmetic crossing conjecture for split links

Joshua Wang

Geometry & Topology 26 (2022) 2941–3053

Given a band sum of a split two-component link along a nontrivial band, we obtain a family of knots indexed by the integers by adding any number of full twists to the band. We show that the knots in this family have the same Heegaard knot Floer homology and the same instanton knot Floer homology. In contrast, a generalization of the cosmetic crossing conjecture predicts that the knots in this family are all distinct. We verify this prediction by showing that any two knots in this family have distinct Khovanov homology. Along the way, we prove that each of the three knot homologies detects the trivial band.

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cosmetic, crossing, nugatory, Khovanov, Floer, instanton, Heegaard Floer, knot Floer, band sum, split links, detection
Mathematical Subject Classification
Primary: 57K10, 57K18
Received: 24 July 2020
Revised: 2 July 2021
Accepted: 10 August 2021
Published: 23 January 2023
Proposed: Ciprian Manolescu
Seconded: Paul Seidel, Tomasz Mrowka
Joshua Wang
Department of Mathematics
Harvard University
Cambridge, MA
United States