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

Abstract

We revisit Rozansky’s construction of Khovanov homology for links in S2×S1, extending it to define the Khovanov homology Kh(L) for links L in Mr=#r(S2×S1) 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 S(Mr) of Hoste and Przytycki when L is null-homologous in Mr. The construction also allows for a clear path toward defining a Lee’s homology Kh′(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 S3 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.