A refinement of Khovanov homology

Lobb, Andrew; Watson, Liam

doi:10.48550/arxiv.1908.00082

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2020

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“…For example, a prominent open problem was resolved when Piccirillo showed that the Conway knot is not slice, using the s-invariant [44] defined by Rasmussen from the spectral sequence from Khovanov homology to Lee homology. Lobb-Watson's [34] filtered invariant is able to detect mutants in the presence of an involution. In a different direction, one may also consider generalized mutations along genus 2 surfaces from which (Conway) mutation may be recovered.…”

Section: 2mentioning

confidence: 99%

On Khovanov Homology and Related Invariants

Caprau¹,

González²,

Lee³

et al. 2020

Preprint

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This paper begins with a survey of some applications of Khovanov homology to lowdimensional topology, with an eye toward extending these results to sl(n) homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to sl(n) homology for n ≥ 2, including the universal sl(2) and sl(3) homology theories. Inspired by Alishahi and Dowlin's bounds for the unknotting number coming from Khovanov homology and relying on spectral sequence arguments, we produce bounds on the alternation number of a knot. Lee and Bar-Natan spectral sequences also provide lower bounds on Turaev genus.

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Section: 2mentioning

confidence: 99%