It provides lots of computable invariants in contact geometry, in particular the <a href="http://arxiv.org/abs/math/0210127">contact invariant</a> defined by Ozsvåth and Szabó via open books and the Giroux correspondence.  For example, on Seifert fibered spaces the question of whether tight contact structures exist was <a href="http://arxiv.org/abs/0709.0737">completely solved</a> by Lisca and Stipsicz, and the classification was completed in many of these cases in several other papers; and Ghiggini used it to exhibit contact structures which are <a href="http://arxiv.org/abs/math/0506380">strongly fillable but not Stein fillable</a>.


Similarly, the <a href="http://arxiv.org/abs/0802.0628">LOSS invariant</a> (named after Lisca-Ozsváth-Stipsicz-Szabó) and the related, easily <a href="http://math.mit.edu/~petero/transverse.html">computable</a> Ozsváth-Szabó-Thurston <a href="http://arxiv.org/abs/math/0611841">grid diagram invariants</a> of Legendrian and transverse knots were used by Ng-Ozsváth-Thurston to successfully <a href="http://arxiv.org/abs/math/0703446">distinguish many pairs of knots</a> in the standard contact $S^3$, and been used to prove other properties such as the fact due to Etnyre and Vela-Vick that the complement of the binding of any open book of any contact structure has <a href="http://arxiv.org/abs/0909.3465">no Giroux torsion</a>.  (According to recent work of Baldwin--Vela-Vick--Vértesi, <a href="http://arxiv.org/abs/1112.5970">these invariants are the same</a> for knots in the standard $S^3$.)