Khovanov homology for links in ${\#}^r(S^2\times S^1)$

Abstract

We revisit Rozansky's construction of Khovanov homology for links in $S^2\times S^1$, extending it to define Khovanov homology $Kh(L)$ for links $L$ in $M^r=#^r(S^2\times S^1)$ for any $r$. The graded Euler characteristic of $Kh(L)$ can be used to recover WRT invariants at certain roots of unity, and also recovers the evaluation of $L$ in the skein module $\mathcal{S}(M^r)$ of Hoste and Przytycki when $L$ is null-homologous in $M^r$. The construction also allows for a clear path towards defining a Lee's homology $K{h}^{\prime }(L)$ 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 $S^3$ 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.

arXiv comment69pgs, many figures. This new version (Oct 22 2019) contains one very important correction, together with several minor updates (thank you to the referee who pointed out many of these)

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Cited by in corpus (2)

• Invariants of 4-manifolds from Khovanov-Rozansky link homology
• Khovanov-Rozansky homology for infinite multi-colored braids