We extend Lipshitz and Sarkar’s definition of a stable homotopy type associated to a link whose cohomology recovers the Khovanov cohomology of . Given an assignment (called a coloring) of a positive integer to each component of a link , we define a stable homotopy type whose cohomology recovers the –colored Khovanov cohomology of . This goes via Rozansky’s definition of a categorified Jones–Wenzl projector as an infinite torus braid on strands.
We then observe that Cooper and Krushkal’s explicit definition of also gives rise to stable homotopy types of colored links (using the restricted palette ), and we show that these coincide with . We use this equivalence to compute the stable homotopy type of the –colored Hopf link and the –colored trefoil. Finally, we discuss the Cooper–Krushkal projector and make a conjecture of for the unknot.
"A Khovanov stable homotopy type for colored links." Algebr. Geom. Topol. 17 (2) 1261 - 1281, 2017. https://doi.org/10.2140/agt.2017.17.1261