The cosmetic crossing conjecture for split links

Wang, Joshua¹

¹ Department of Mathematics, Harvard University, Cambridge, MA, United States

Geometry & topology, Tome 26 (2022) no. 7, pp. 2941-3053.

Résumé

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.

Détail

DOI : 10.2140/gt.2022.26.2941

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

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