  We prove that a nicely fibered link (by which we mean the binding of an open
book) in a tight contact manifold $(M,\xi)$ 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
$\xi.$ This gives a geometric reason for the non-sharpness of the Bennequin
bound for fibered links. We also note that this allows the classification, up
to contactomorphism, of maximal self-linking number links in these knot types.
Moreover, in the standard tight contact structure on $S^3$ we classify, up to
transverse isotopy, transverse knots with maximal self-linking number in the
knots 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-postitive braids are
related by positive Markov stabilizations and when a minimal braid index
representative of a knot is quasi-positive.
  In the new version we also prove 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.
