Contact Homology of Orbit Complements and Implied Existence

TL;DR

This paper uses cylindrical contact homology to relate the existence of certain Reeb orbits with specific knotting properties to the existence of other orbits with different knotting properties, with applications to geodesics on spheres.

Contribution

It introduces a method to infer the existence of Reeb orbits with different knotting/linking properties based on known orbits, advancing the understanding of contact homology in 3-manifolds.

Findings

Examples on the 3-sphere illustrating the theory

Application to closed geodesics on S^2

Relation between knotting properties of Reeb orbits

Abstract

For Reeb vector fields on closed 3-manifolds, cylindrical contact homology is used to show that the existence of a set of closed Reeb orbit with certain knotting/linking properties implies the existence of other Reeb orbits with other knotting/linking properties relative to the original set. We work out a few examples on the 3-sphere to illustrate the theory, and describe an application to closed geodesics on $S^2$ (a version of a result due to Angenent).