Planar open books and Floer homology
  Giroux has described a correspondence between open book decompositions on a 3--manifold and contact structures. In this paper we use Heegaard Floer homology to give restrictions on contact structures which correspond to open book decompositions with planar pages, generalizing a recent result of Etnyre. 
Peter Ozsvath, Andras I. Stipsicz, Zoltan Szabo