Mirror links have dual odd and generalized Khovanov homology. (English) Zbl 1410.57011

Summary: We show that the generalized Khovanov homology defined by the first author in the framework of chronological cobordisms [K. Putyra, “Cobordisms with chronologies and a generalisation of the Khovanov complex”, Preprint, arXiv:1004.0889 and Banach Cent. Publ. 103, 291–355 (2014; Zbl 1336.57024)] admits a grading by the group \({\mathbb{Z}\times\mathbb{Z}_2}\), in which all homogeneous summands are isomorphic to the unified Khovanov homology defined over the ring \(\mathbb{Z}_{\pi}:=\mathbb{Z}[\pi]/(\pi^2-1)\). (Here, setting \(\pi\) to \(\pm 1\) results either in even or odd Khovanov homology.) The generalized homology has \(\Bbbk:=\mathbb{Z}[X,Y,Z^{\pm1}]/(X^2{=}Y^2{=}1)\) as coefficients, and the above implies that most automorphisms of \(\Bbbk\) fix the isomorphism class of the generalized homology regarded as a \(\Bbbk\)-module, so that the even and odd Khovanov homology are the only two specializations of the invariant. In particular, switching \(X\) with \(Y\) induces a derived isomorphism between the generalized Khovanov homology of a link \(L\) with its dual version, i.e. the homology of the mirror image \(L^!\), and we compute an explicit formula for this map. When specialized to integers it descends to a duality isomorphism for odd Khovanov homology, which was conjectured by A. N. Shumakovitch [J. Knot Theory Ramifications 20, No. 1, 203–222 (2011; Zbl 1223.57013)].


57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
55N35 Other homology theories in algebraic topology
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