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Khovanov homotopy calculations using flow category calculus

Lobb, Andrew; Orson, Patrick; Schuetz, Dirk

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Authors

Patrick Orson



Abstract

The Lipshitz–Sarkar stable homotopy link invariant defines Steenrod squares on the Khovanov cohomology of a link. Lipshitz–Sarkar constructed an algorithm for computing the first two Steenrod squares. We develop a new algorithm which implements the flow category simplification techniques previously defined by the authors and Dan Jones. We give a purely combinatorial approach to calculating the second Steenrod square and Bockstein homomorphisms in Khovanov cohomology, and flow categories in general. The new method has been implemented in a computer program by the third author and applied to large classes of knots and links. Several homotopy types not previously witnessed are observed, and more evidence is obtained that Khovanov stable homotopy types do not contain CP2 as a wedge summand. In fact, we are led by our calculations to formulate an even stronger conjecture in terms of Z/2 summands of the cohomology.

Citation

Lobb, A., Orson, P., & Schuetz, D. (2020). Khovanov homotopy calculations using flow category calculus. Experimental Mathematics, 29(4), 475-500. https://doi.org/10.1080/10586458.2018.1482805

Journal Article Type Article
Acceptance Date May 29, 2018
Online Publication Date Apr 9, 2019
Publication Date 2020
Deposit Date Jun 1, 2018
Publicly Available Date Apr 9, 2020
Journal Experimental Mathematics
Print ISSN 1058-6458
Electronic ISSN 1944-950X
Publisher Taylor and Francis Group
Peer Reviewed Peer Reviewed
Volume 29
Issue 4
Pages 475-500
DOI https://doi.org/10.1080/10586458.2018.1482805

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