
Knot Types of Periodic Reeb Orbits and Their Role in 4-Dimensional Symplectic Topology
Umberto Hryniewicz of RWTH Aachen presents a research seminar talk for the IAS symplectic geometry zoominar, drawing on joint work with Pedro Salomao and Richard Siefring, and separately with Michael Hutchings and Vinicius Ramos. He examines how dynamical convexity restricts the knot types of periodic Reeb orbits in 4-dimensional symplectic topology. The core result concerns dynamically convex star-shaped domains in a 4-dimensional symplectic vector space, where the minimal action among unknotted Hopf orbits with self-linking number -1 satisfies the axioms of a normalized symplectic capacity. Combined with a result of Edtmair, this shows the quantity equals the cylindrical capacity, and Hryniewicz explains why this in turn equals the first ECH capacity without invoking Seiberg-Witten theory. He closes by discussing transverse knot types that cannot appear as periodic Reeb orbits of a dynamically convex contact form on the standard contact 3-sphere.