Neighborhoods of transverse knots and destabilizations

TL;DR

This paper establishes the uniqueness of neighborhoods for transverse knots, proves a structure theorem for non-loose Legendrian knots, and demonstrates a finiteness result for transverse knots in tight contact manifolds.

Contribution

It introduces a general destabilization theorem for Legendrian knots and reveals a manifold with infinitely many tight contact structures without Giroux torsion.

Findings

Transverse knots have unique standard neighborhoods.

A structure theorem for non-loose Legendrian knots is proved.

Finiteness of transverse knots in certain tight contact manifolds is established.

Abstract

In this note, we show that transverse knots have unique standard neighborhoods and prove a structure theorem about non-loose Legendrian knots. We also prove a finiteness result for transverse knots in a tight contact manifold. The common theme of these two results is a general destabilization result for Legendrian knots. As a byproduct of this work, we find a manifold with an infinite number of distinct tight contact structures, up to contactomoprhism, with no Giroux torsion.