Algebraic & Geometric Topology

A Khovanov stable homotopy type for colored links

Andrew Lobb, Patrick Orson, and Dirk Schütz

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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.

Article information

Algebr. Geom. Topol., Volume 17, Number 2 (2017), 1261-1281.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds

Khovanov flow category stable homotopy type


Lobb, Andrew; Orson, Patrick; Schütz, Dirk. A Khovanov stable homotopy type for colored links. Algebr. Geom. Topol. 17 (2017), no. 2, 1261--1281. doi:10.2140/agt.2017.17.1261.

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