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

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.

32 pages, with corrections and updated references. The construction of the pseudo-holomorphic plane from a symplectically embedded disk is now done starting with a transverse knot. We thank Shah Faisal for spotting a mistake in the statement Theorem 1.4 of a previous version of this paper: the periodic orbit claimed to exist in the concave end is not necessarily contractible