A Bar-Natan homotopy type

Abstract

A spatial refinement of Bar-Natan homology is given, that is, for any link diagram D we construct a CW-spectrum BN(D) whose reduced cellular cochain complex gives the Bar-Natan complex of D. The stable homotopy type of BN(D) is a link invariant and is described as the wedge sum of the "canonical sphere spectra". We conjecture that the quantum filtration of Bar-Natan homology also lifts to the spatial level, and that it leads us to a cohomotopical refinement of the s-invariant.