2018/01/02 by Cavallo, Alberto
#57K18 #57K33 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1801.00614
We prove that a version of the Thurston-Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere (M,ξ), whenever ξ is tight. More specifically, we show that the self-linking number of a transverse link T in (M,ξ), such that the boundary of its tubular neighbourhood consists of incompressible tori, is bounded by the Thurston norm ||T||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 S3. 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.