On Legendrian representatives of non-fibered knots

TL;DR

This paper investigates Legendrian representatives of non-fibered knots in contact 3-manifolds, establishing conditions under which such knots can have Legendrian representatives with specific Thurston-Bennequin invariants, extending previous results beyond the 3-sphere.

Contribution

It proves that non-trivial knots realizing the Thurston-Bennequin bound have Legendrian representatives with tb=0, generalizing to contact manifolds with unique contact invariants, and relates tau invariants to fiberedness.

Findings

Non-trivial knots with maximal Thurston-Bennequin bound have Legendrian representatives with tb=0.

The result extends to contact manifolds with unique contact invariants.

For nearly fibered knots, tau equals genus implies realization of the Thurston-Bennequin bound.

Abstract

We show that in $(S^3,\xi_{std})$ if $K$ is a non-trivial knot that realizes the three-dimensional Thurston-Bennequin bound (i.e. $K$ has a Legendrian representative $\Lambda$ with $tb(\Lambda)-rot(\Lambda)=2g(K)-1$), then $K$ has a Legendrian representative $L$ with $tb=0$. Moreover, this result can be easily generalized to contact manifolds that uniquely represent the associated contact invariants. This is the first result on Legendrian representatives of non-fibered knots in $3-$manifolds other than $S^3$. We also show that if $K$ is a nearly fibered knot in $S^3$ then $\tau(K)=g(K)$ implies that $K$ realizes the three-dimensional Thurston-Bennequin bound.