Unknotted Reeb orbits and nicely embedded holomorphic curves

TL;DR

This paper demonstrates that certain contact 3-manifolds admit unknotted Reeb orbits with self-linking number -1, using holomorphic curve intersection theory to extend previous results and introduce new techniques in symplectic topology.

Contribution

It introduces a novel approach using intersection theory of holomorphic curves to identify unknotted Reeb orbits in low-dimensional contact manifolds, generalizing prior theorems.

Findings

Existence of unknotted Reeb orbits with self-linking -1 in specific contact 3-manifolds

Development of a local adjunction formula for holomorphic annuli

Application of intersection theory to understand holomorphic curve compactification

Abstract

We exhibit a distinctly low-dimensional dynamical obstruction to the existence of Liouville cobordisms: for any contact 3-manifold admitting an exact symplectic cobordism to the tight 3-sphere, every nondegenerate contact form admits an embedded Reeb orbit that is unknotted and has self-linking number -1. The same is true moreover for any contact structure on a closed 3-manifold that is reducible. Our results generalize an earlier theorem of Hofer-Wysocki-Zehnder for the 3-sphere, but use somewhat newer techniques: the main idea is to exploit the intersection theory of punctured holomorphic curves in order to understand the compactification of the space of so-called "nicely embedded" curves in symplectic cobordisms. In the process, we prove a local adjunction formula for holomorphic annuli breaking along a Reeb orbit, which may be of independent interest.