A Khovanov stable homotopy type for colored links

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 X-col(L-c) 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 {1,2}),and we show that these coincide with X-col. 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 P-3 and make a conjecture of X-col(U-3) for U the unknot.