Khovanov homology, open books, and tight contact structures

We define the reduced Khovanov homology of an open book (S,ϕ), and identify a distinguished “contact element” in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,ϕ). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Stein-fillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology of an open book which parallels the comultiplication on Heegaard Floer homology defined in Baldwin (2008) [4].