Khovanov homology for links in \(\#^r (S^2\times S^1)\)
In this paper the author generalises Khovanov homology to links in a \(3\)-sphere with handles attached. When \(r=0\), the manifold \(M^r = S^3\) and the author recovers classical Khovanov homology. When \(r=1\), \(M^r = S^2\times S^1\) and the author recovers the Khovanov homology for links in \(S^2\times S^1\) of \textit{L. Rozansky} [``A categorification of the stable SU(2) Witten-Reshetikhin-Turaev invariant of links in \(S^2\times S^1\)'', Preprint, \url{arXiv:1011.1958}]. The \(3\)-sphere with \(r\) handles \(M^r\) can be visualised as the complement in \(S^3\) of \(r\) pairs of balls, where the boundary of each ball is identified with the boundary of the other ball in its pair. Now fix a segment in the boundary of each ball, so that the two segments in each pair of balls are identified. Fix also an embedding of a square \([0,1]^2\) in \(S^3\) (without removing the balls) that isotopes each segment into its paired segment. A link in \(M^r\) can be represented as a tangle in that complement disjoint from the embedded square whose intersection with the boundary of each ball lies in the segment and has the same cardinality as the intersection with the boundary of the other ball in the pair. To each such representation of a link \(L\) one can associate a sequence of links \(\{L(k)_{k\geq 0}\}\) in \(S^3\) as follows: \(L(0)\) is the result of closing the tangle by gluing parallel strands in each square. \(L(k)\) is the result of performing \(k\) full twists to the parallel strands in each square of \(L(0)\). After appropriate shiftings and homotopy replacements whose definition takes two fifths of the paper, the Khovanov chain complexes of the links \(\{L(k)\}_{k\geq 0}\) form a directed system and the Khovanov chain complex of the link \(L\) is its colimit. The Khovanov homology of the link \(L\) is the homology of this colimit, which is shown to be a link invariant and functorial with respect to cobordisms up to sign. These constructions apply only if the intersection number of the link with each of the belt spheres of the handles of \(M^r\) is even, otherwise the Khovanov homology of the link is defined to be trivial. The graded Euler characteristic of the Khovanov homology of a link in \(M^r\) recovers the Witten-Reshetikhin-Turaev invariant of the link. The relation with the skein module of \(M^r\) of [\textit{J. Hoste} and \textit{J. H. Przytycki}, in: Knots 90. Proceedings of the international conference on knot theory and related topics, held in Osaka, Japan, August 15-19, 1990. Berlin etc.: Walter de Gruyter. 363--379 (1992; Zbl 0772.57022)] is also discused. Finally, there is a construction that associates to each isotopy class of links in \(S^3\) an isotopy class of knots in \(M^r\) called \textit{knotification}. The author also provides a direct method for computing the Khovanov homology of the knotification of any link in a single step.
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