A Khovanov stable homotopy type for colored links

TL;DR

This paper extends the stable homotopy type framework for links to include colored links, connecting Khovanov cohomology with Rozansky's categorified Jones-Wenzl projectors, and computes specific examples.

Contribution

It introduces a new stable homotopy type for colored links based on Rozansky's projectors, unifying existing constructions and enabling explicit computations.

Findings

Defined stable homotopy types for colored links using Rozansky's projectors

Established equivalence with Cooper-Krushkal's projectors for certain colors

Computed stable homotopy types for specific colored links like the Hopf link and trefoil

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.