A combinatorial description of the $U^2=0$ version of Heegaard Floer   homology
  We show that every 3--manifold admits a Heegaard diagram in which a truncated version of Heegaard Floer homology (when the holomorpic disks pass through the basepoints at most once) can be computed combinatorially. 
Peter Ozsvath, Andras Stipsicz, Zoltan Szabo