Fibered Transverse Knots and the Bennequin Bound

Abstract

We prove that a nicely fibered link (by which we mean the binding of an open book) in a tight contact manifold (M, ξ) with zero Giroux torsion has a transverse representative realizing the Bennequin bound if and only if the contact structure it supports (since it is also the binding of an open book) is ξ. This gives a geometric reason for the non-sharpness of the Bennequin bound for fibered links. We also give the classification, up to contactomorphism, of maximal self-linking number links in these knot types. Moreover, in the standard tight contact structure on S3 we classify, up to transverse isotopy, transverse knots with maximal self-linking number in the knot types given as closures of positive braids and given as fibered strongly quasi-positive knots. We derive several braid theoretic corollaries from this. In particular, we give conditions under which quasi-positive braids are related by positive Markov stabilizations and when a minimal braid index representative of a knot is quasi-positive. Finally, we observe that our main result can be used to show, and make rigorous the statement, that contact structures on a given manifold are in a strong sense classified by the transverse knot theory they support.