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2017 A Khovanov stable homotopy type for colored links
Andrew Lobb, Patrick Orson, Dirk Schütz
Algebr. Geom. Topol. 17(2): 1261-1281 (2017). DOI: 10.2140/agt.2017.17.1261

Abstract

We extend Lipshitz and 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 a positive integer to each component of a link L, we define a stable homotopy type Xcol(Lc) whose cohomology recovers the c–colored Khovanov cohomology of L. This goes via Rozansky’s definition of a categorified Jones–Wenzl projector Pn as an infinite torus braid on n strands.

We then observe that Cooper and Krushkal’s explicit definition of P2 also gives rise to stable homotopy types of colored links (using the restricted palette {1,2}), and we show that these coincide with Xcol. 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 P3 and make a conjecture of Xcol(U3) for U the unknot.

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Andrew Lobb. Patrick Orson. Dirk Schütz. "A Khovanov stable homotopy type for colored links." Algebr. Geom. Topol. 17 (2) 1261 - 1281, 2017. https://doi.org/10.2140/agt.2017.17.1261

Information

Received: 27 April 2016; Revised: 12 August 2016; Accepted: 21 August 2016; Published: 2017
First available in Project Euclid: 19 October 2017

zbMATH: 1375.57019
MathSciNet: MR3623688
Digital Object Identifier: 10.2140/agt.2017.17.1261

Subjects:
Primary: 57M27

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.17 • No. 2 • 2017
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