Knot types of periodic Reeb orbits and their role in 4-dimensional symplectic topology

This talk, which is based on two joint works, one with Pedro Salomão and Richard Siefring and another with Michael Hutchings and Vinicius Ramos, revolves around the role that restrictions on the knot types of periodic Reeb orbits imposed by the assumption of dynamical convexity plays in 4-dimensional symplectic topology. For instance, for dynamically convex star-shaped domains in a 4-dimensional symplectic vector space, the minimal action among periodic Reeb orbits in the boundary which are unknotted and have self-linking number -1, called Hopf orbits, satisfies the axioms of a normalized symplectic capacity. This shows that this number is not larger than the cylindrical capacity. A result of Edtmair establishes the other inequality, and these two results combined yield that the minimal action of a Hopf orbit is equal to the cylindrical capacity of such a domain. We will discuss why is this equal to the first ECH capacity, which then explains the latter capacity in simple and purely symplectic geometric terms, with no need of Seiberg-Witten theory. We will also discuss why several transverse knot types in the standard contact 3-sphere cannot be realized as periodic Reeb orbits of a dynamically convex contact form.