Volume 17, issue 2 (2017)

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A Khovanov stable homotopy type for colored links

Andrew Lobb, Patrick Orson and Dirk Schütz

Algebraic & Geometric Topology 17 (2017) 1261–1281
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 ${\mathsc{X}}_{col}\left({L}_{c}\right)$ 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 and Krushkal’s explicit definition of ${P}_{2}$ also gives rise to stable homotopy types of colored links (using the restricted palette $\left\{1,2\right\}$), and we show that these coincide with ${\mathsc{X}}_{col}$. We use this equivalence to compute the stable homotopy type of the $\left(2,1\right)$–colored Hopf link and the $2$–colored trefoil. Finally, we discuss the Cooper–Krushkal projector ${P}_{3}$ and make a conjecture of ${\mathsc{X}}_{col}\left({U}_{3}\right)$ for $U$ the unknot.

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