A Khovanov stable homotopy type for colored links

Lobb, Andrew; Orson, Patrick; Schuetz, Dirk

doi:10.2140/agt.2017.17.1261

A Khovanov stable homotopy type for colored links

Lobb, Andrew; Orson, Patrick; Schuetz, Dirk

Authors

Professor Andrew Lobb andrew.lobb@durham.ac.uk
Professor

Patrick Orson

Professor Dirk Schuetz dirk.schuetz@durham.ac.uk
Professor

Abstract

We extend Lipshitz-Sarkar's definition of a stable homotopy type associated to a link L whose cohomology recovers the Khovanov cohomology of L. Given an assignment c (called a coloring) of positive integer to each component of a link L, we define a stable homotopy type X_col(L_c) whose cohomology recovers the c-colored Khovanov cohomology of L. This goes via Rozansky's definition of a categorified Jones-Wenzl projector P_n as an infinite torus braid on n strands. We then observe that Cooper-Krushkal's explicit definition of P_2 also gives rise to stable homotopy types of colored links (using the restricted palette {1, 2}), and we show that these coincide with X_col. We use this equivalence to compute the stable homotopy type of the (2,1)-colored Hopf link and the 2-colored trefoil. Finally, we discuss the Cooper-Krushkal projector P_3 and make a conjecture of X_col(U_3) for U the unknot.

Citation

Lobb, A., Orson, P., & Schuetz, D. (2017). A Khovanov stable homotopy type for colored links. Algebraic & geometric topology, 17(2), 1261-1281. https://doi.org/10.2140/agt.2017.17.1261

Journal Article Type	Article
Acceptance Date	Aug 21, 2016
Online Publication Date	Mar 14, 2017
Publication Date	Mar 14, 2017
Deposit Date	Aug 22, 2016
Publicly Available Date	Apr 21, 2017
Journal	Algebraic and Geometric Topology
Print ISSN	1472-2747
Electronic ISSN	1472-2739
Publisher	Mathematical Sciences Publishers (MSP)
Peer Reviewed	Peer Reviewed
Volume	17
Issue	2
Pages	1261-1281
DOI	https://doi.org/10.2140/agt.2017.17.1261

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Copyright Statement
First published in Algebraic & Geometric Topology in 17 (2017) 1261–1281, published by Mathematical Sciences Publishers. © 2017 Mathematical Sciences Publishers. All rights reserved.