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Khovanov homology for virtual knots with arbitrary coefficients. (English) Zbl 1117.57013

In previous papers, the author constructed the Khovanov theory for virtual knots, but the Khovanov complex was well defined for all virtual links only over \(\mathbb{Z}_2\), and the Khovanov homology with arbitrary coefficients was defined only for virtual knots corresponding to orientable atoms. In the present paper, the Khovanov complex with arbitrary coefficients for arbitrary virtual knots is constructed. Key ideas to overcome the difficulty are the change of basis of the Frobenius algebra representing the Khovanov homology of the unknot and the exterior product instead of the usual tensor product for the spaces corresponding to the circles. The method above works for twisted virtual knots in the sense of Bourgoin and Viro.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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