Symplectic dynamics and the 3-sphere

TL;DR

This paper proves that if a knot's exterior admits a special type of contact form, then the 3-manifold must be a 3-sphere and the knot is unknotted, linking symplectic dynamics to knot theory.

Contribution

It establishes a novel connection between symplectic dynamics and the topology of knots in 3-manifolds, characterizing the 3-sphere via contact forms.

Findings

The exterior of a knot with an aperiodic contact form implies the manifold is a 3-sphere.

Under these conditions, the knot must be the unknot.

The result links contact geometry to classical knot theory.

Abstract

Given a knot in a closed connected orientable 3-manifold we prove that if the exterior of the knot admits an aperiodic contact form that is Euclidean near the boundary, then the 3-manifold is diffeomorphic to the 3-sphere and the knot is the unknot.