Applications of Ozsvath-Szabo invariants to contact geometry

Let T ⊂ (Y, xi) be a transverse knot which is the binding of some open book, (T, pi), for the ambient contact manifold (Y, xi). We show that the transverse invariant J&d14; T∈HFK&d14; (-Y, K), defined in [LOSS08], is nonvanishing for such transverse knots. We also prove a vanishing theorem for the invariants L and J . As a corollary of these two facts, we see that if (T, pi) is an open book with connected binding, then the complement of T has no Giroux torsion.;More generally, we prove using different methods that if (B, pi) is any open book decomposition of (Y, xi), then the complement of B is torsion-free.;We also show by explicit computation that the sutured Floer contact invariant can distinguish isotopy classes of tight contact structures on solid tori with convex boundary and 2n vertical dividing curves.