Volume 24, issue 2 (2020)

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Khovanov homotopy type, Burnside category and products

Tyler Lawson, Robert Lipshitz and Sucharit Sarkar

Geometry & Topology 24 (2020) 623–745
Abstract

We give a new construction of a Khovanov stable homotopy type, or spectrum. We show that this construction gives a space stably homotopy equivalent to the Khovanov spectra constructed by Lipshitz and Sarkar  (J. Amer. Math. Soc. 27 (2014) 983–1042) and Hu, Kriz and Kriz (Topology Proc. 48 (2016) 327–360) and, as a corollary, that those two constructions give equivalent spectra. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying several of Lipshitz and Sarkar’s conjectures. Finally, combining these results with Lipshitz and Sarkar’s computations (J. Topol. 7 (2014) 817–848) and refined $s$–invariant (Duke Math. J. 163 (2014) 923–952), we obtain new results about the slice genera of certain knots.

Keywords
Khovanov homotopy, Khovanov homology, flow categories
Mathematical Subject Classification 2010
Primary: 55P42, 57M25
Publication
Received: 12 May 2015
Revised: 8 January 2019
Accepted: 20 August 2019
Published: 23 September 2020
Proposed: Ciprian Manolescu
Seconded: Ralph Cohen, Ian Agol
Authors
 Tyler Lawson Department of Mathematics University of Minnesota Minneapolis, MN United States Robert Lipshitz Department of Mathematics University of Oregon Eugene, OR United States Sucharit Sarkar Department of Mathematics University of California at Los Angeles Los Angeles, CA United States