A Khovanov stable homotopy type for colored links

Abstract: We extend Lipshitz-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 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-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.