Cobordisms with chronologies and a generalisation of the Khovanov complex

There are two categorifications of the Jones polynomial: "even" discovered by M.Khovanov in 1999 and "odd" dicovered by P.Ozsvath, J.Rasmussen and Z.Szabo in 2007. The first one can be fully constructed in the category of cobordisms (strictly: in the additive closure of that category), where we can build a complex for a given tangle and show its invariance under Reidemeister moves. The even link homology is given by a monoidal functor from cobordisms into modules. However, odd link homology cannot be obtained in this way. In this paper I fill this gap. I enrich cobordisms with chronologies (projections onto intervals which are Morse separable functions) and show that they form a category. Given a tangle diagram I take the cube of its resolutions and build a complex in this new category. I show this complex is a tangle invariant and applying appropriate functors I can recover both even and odd link homology theories.

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