Given a contact structure on a closed, oriented three-manifold , we describe an invariant that takes values in the three-manifold's Floer homology . This invariant vanishes for overtwisted contact structures and is nonzero for Stein-fillable ones. The construction uses Giroux's interpretation of contact structures in terms of open-book decompositions.
"Heegaard Floer homology and contact structures." Duke Math. J. 129 (1) 39 - 61, 15 July 2005. https://doi.org/10.1215/S0012-7094-04-12912-4