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 ${\mathcal{X}}_{col}\left(L_c\right)$ 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 ${\mathcal{X}}_{col}$. 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 ${\mathcal{X}}_{col}\left(U_3\right)$ for $U$ the unknot.