A Bar-Natan homotopy type

Taketo Sano

doi:10.1142/s0129167x23500088

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arXiv · 2102.07529

A Bar-Natan homotopy type

Taketo Sano

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Abstract

A spatial refinement of Bar-Natan homology is given, that is, for any link diagram $D$ we construct a CW-spectrum $\mathcal{X}_{\mathit{BN}}(D)$ whose reduced cellular cochain complex gives the Bar-Natan complex of $D$. The stable homotopy type of $\mathcal{X}_{\mathit{BN}}(D)$ 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.

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Taketo Sano. 2021-12-28. A Bar-Natan homotopy type. https://doi.org/10.1142/s0129167x23500088

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