On Khovanov Homology and Related Invariants

This paper begins with a survey of some applications of Khovanov homology to low-dimensional topology, with an eye toward extending these results to \(\mathfrak{sl}(n)\) homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to \(\mathfrak{sl}(n)\) homolo...

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Bibliographic Details
Published in:arXiv.org 2020-02
Main Authors: Caprau, Carmen, González, Nicolle, Christine Ruey Shan Lee, Lowrance, Adam M, Sazdanović, Radmila, Zhang, Melissa
Format: Article
Language:English
Subjects:
Homology
Lower bounds
Sequences
Topology
Online Access:Get full text
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Summary:This paper begins with a survey of some applications of Khovanov homology to low-dimensional topology, with an eye toward extending these results to \(\mathfrak{sl}(n)\) homologies. We extend Levine and Zemke's ribbon concordance obstruction from Khovanov homology to \(\mathfrak{sl}(n)\) homology for \(n \geq 2\), including the universal \(\mathfrak{sl}(2)\) and \(\mathfrak{sl}(3)\) 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.
ISSN:2331-8422