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Khovanov homotopy calculations using flow category calculus.

Lobb, Andrew and Orson, Patrick and Schuetz, Dirk (2020) 'Khovanov homotopy calculations using flow category calculus.', Experimental mathematics., 29 (4). pp. 475-500.

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.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1080/10586458.2018.1482805
Publisher statement:This is an Accepted Manuscript of an article published by Taylor & Francis in Experimental Mathematics on 09 April 2019 available online: http://www.tandfonline.com/10.1080/10586458.2018.1482805
Date accepted:29 May 2018
Date deposited:12 June 2018
Date of first online publication:09 April 2019
Date first made open access:09 April 2020

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