Fibered Transverse Knots and the Bennequin Bound

TL;DR

This paper characterizes when fibered links in tight contact manifolds realize the Bennequin bound, classifies maximal self-linking number knots in certain types, and explores implications for contact structure classification.

Contribution

It provides a geometric criterion for the Bennequin bound realization, classifies maximal self-linking knots, and connects transverse knot theory with contact structure classification.

Findings

Characterizes when fibered links realize the Bennequin bound.

Classifies maximal self-linking number knots in specific knot types.

Establishes conditions for quasi-positive braids related by positive Markov stabilizations.

Abstract

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.