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 ${\mathcal{X}}_{BN}(D)$ whose reduced cellular cochain complex gives the Bar-Natan complex of $D$. The stable homotopy type of ${\mathcal{X}}_{BN}(D)$ is a link invariant and is described as the wedge sum of the canonical cells. 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.

49 pages

References in corpus (2)

• Computations of the Lipshitz-Sarkar Steenrod Square on Khovanov Homology
• Fixing the functoriality of Khovanov homology: a simple approach