On Bennequin-type inequalities for links in tight contact 3-manifolds

We prove that a version of the Thurston–Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere [Formula: see text], whenever [Formula: see text] is tight. More specifically, we show that the self-linking number of a transverse link [Formula: see text] in [Formula: see text], such that the boundary of its tubular neighborhood consists of incompressible tori, is bounded by the Thurston norm [Formula: see text] of [Formula: see text]. A similar inequality is given for Legendrian links by using the notions of positive and negative transverse push-off. We apply this bound to compute the tau-invariant for every strongly quasi-positive link in [Formula: see text]. This is done by proving that our inequality is sharp for this family of smooth links. Moreover, we use a stronger Bennequin inequality, for links in the tight 3-sphere, to generalize this result to quasi-positive links and determine their maximal self-linking number.