Khovanov homology, open books, and tight contact structures

Abstract

We define the reduced Khovanov homology of an open book (S,h), and we 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,h). 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 previously by the first author.

arXiv commentAdded Proposition 1.4 regarding non-fillability. Expanded Lemma 5.1 and corrected its proof

References in corpus (9)

• Floer homology and knot complements
• On the Khovanov and knot Floer homologies of quasi-alternating links
• Heegaard Floer homologies and contact structures
• Khovanov homology and the slice genus
• On arc index and maximal Thurston-Bennequin number
• On the contact class in Heegaard Floer homology
• Comultiplicativity of the Ozsvath-Szabo contact invariant
• Contact Structures with Distinct Heegaard Floer Invariants
• On Transverse Knots and Branched Covers

Cited by in corpus (4)

• Khovanov homology, sutured Floer homology, and annular links
• On the spectral sequence from Khovanov homology to Heegaard Floer homology
• Odd Khovanov homology is mutation invariant
• On the naturality of the spectral sequence from Khovanov homology to Heegaard Floer homology