## Algebraic & Geometric Topology

### A Khovanov stable homotopy type for colored links

#### 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.

#### Article information

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

Dates
Revised: 12 August 2016
Accepted: 21 August 2016
First available in Project Euclid: 19 October 2017

https://projecteuclid.org/euclid.agt/1508431460

Digital Object Identifier
doi:10.2140/agt.2017.17.1261

Mathematical Reviews number (MathSciNet)
MR3623688

Zentralblatt MATH identifier
1375.57019

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

#### Citation

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. https://projecteuclid.org/euclid.agt/1508431460

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