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 $(M,\xi )$, whenever $\xi$ is tight. More specifically, we show that the self-linking number of a transverse link $T$ in $(M,\xi )$, such that the boundary of its tubular neighbourhood consists of incompressible tori, is bounded by the Thurston norm $\mathrm{\mid }\mathrm{\mid }T\mathrm{\mid }{\mathrm{\mid }}_T$ of $T$. 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 $S^3$. 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.