The cosmetic crossing conjecture for split links

Wang, Joshua

doi:10.2140/gt.2022.26.2941

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The cosmetic crossing conjecture for split links

Joshua Wang

Geometry & Topology 26 (2022) 2941–3053

DOI: 10.2140/gt.2022.26.2941

Abstract

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.

Keywords

cosmetic, crossing, nugatory, Khovanov, Floer, instanton, Heegaard Floer, knot Floer, band sum, split links, detection

Mathematical Subject Classification

Primary: 57K10, 57K18

References

Publication

Received: 24 July 2020

Revised: 2 July 2021

Accepted: 10 August 2021

Published: 23 January 2023

Proposed: Ciprian Manolescu

Seconded: Paul Seidel, Tomasz Mrowka

Authors

Joshua Wang
Department of Mathematics Harvard University Cambridge, MA United States

Volume 26, issue 7 (2022)