Khovanov Homology for Links in #r(S2×S1)

Abstract

We revisit Rozansky’s construction of Khovanov homology for links in ${\mathit{S}}^2\times {\mathit{S}}^1$, extending it to define the Khovanov homology $\mathrm{Kh}(\mathit{L})$ for links L in ${\mathit{M}}^{\mathit{r}}={\mathrm{\#}}^{\mathit{r}}({\mathit{S}}^2\times {\mathit{S}}^1)$ for any r. The graded Euler characteristic of $\mathrm{Kh}(\mathit{L})$ can be used to recover WRT invariants at certain roots of unity and also recovers the evaluation of L in the skein module $\mathcal{S}({\mathit{M}}^{\mathit{r}})$ of Hoste and Przytycki when L is null-homologous in ${\mathit{M}}^{\mathit{r}}$. The construction also allows for a clear path toward defining a Lee’s homology ${\mathrm{Kh}}^{\prime }(\mathit{L})$ and associated s-invariant for such L, which we will explore in an upcoming paper. We also give an equivalent construction for the Khovanov homology of the knotification of a link in ${\mathit{S}}^3$ and show directly that this is invariant under handle-slides, in the hope of lifting this version to give a stable homotopy type for such knotifications in a future paper.