Khovanov homology of positive links and of L-space knots

Abstract

We determine the structure of the Khovanov homology groups in homological grading 1 of positive links. More concretely, we show that the first Khovanov homology is supported in a single quantum grading determined by the Seifert genus of the link, where the group is free abelian and of rank determined by the Seifert graph of any of its positive link diagrams. In particular, for a positive link, the first Khovanov homology is vanishing if and only if the link is fibered. Moreover, we extend these results to (p,q)-cables of positive knots whenever $q\geq p$. We also show that several infinite families of Heegaard Floer L-space knots have vanishing first Khovanov homology. This suggests a possible extension of our results to L-space knots.