Khovanov homotopy calculations using flow category calculus

Lobb, Andrew; Orson, Patrick; Schuetz, Dirk

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
• Public URL: https://durham-repository.worktribe.com/output/1329654