On the knot types of periodic Reeb orbits of dynamically convex contact forms

TL;DR

This paper identifies specific transverse knot types on the standard contact 3-sphere that cannot be realized as periodic Reeb orbits of dynamically convex contact forms, highlighting limitations in the types of knots associated with such dynamical systems.

Contribution

It demonstrates the existence of transverse knot types that cannot be realized as periodic Reeb orbits in dynamically convex contact forms, revealing new constraints in contact topology.

Findings

Certain transverse knot types are not realizable as periodic Reeb orbits.

Such knot types do not appear as closed characteristics on convex energy levels.

The results delineate limitations of dynamically convex contact forms in knot realization.

Abstract

We exhibit transverse knot types on the standard contact $3$-sphere that cannot be realized as periodic Reeb orbits of a dynamically convex contact form. In particular, such transverse knot types do not arise as closed characteristics of strictly convex energy levels on a four dimensional symplectic vector space.