Jones–Wenzl projectors and the Khovanov homotopy of the infinite twist

Abstract

We construct and study a lift of Jones–Wenzl projectors to the setting of Khovanov spectra, and provide a realization of such lifted projectors via a Cooper–Krushkal-like sequence of maps. We also give a polynomial action on the 3-strand spectral projector allowing a complete computation of the $3$-colored Khovanov spectrum of the unknot, proving a conjecture of Lobb–Orson–Schütz. As a byproduct, we disprove a conjecture of Lawson–Lipshitz–Sarkar on the topological Hochschild homology of tangle spectra.