Torus knotted Reeb dynamics and the Calabi invariant

Abstract

We establish the existence of a secondary Reeb orbit set with quantitative action and linking bounds for any contact form on the standard tight three-sphere admitting the standard transverse positive $T(p,q)$ torus knot as an elliptic Reeb orbit with a canonically determined rotation number. This can be interpreted through an ergodic lens for Reeb flows transverse to a surface of section. Our results also allow us to deduce an upper bound on the mean action of periodic orbits of naturally associated classes of area preserving diffeomorphisms of the associated Seifert surfaces of genus $(p-1)(q-1)/2$ in terms of the Calabi invariant, without the need for genericity or Hamiltonian hypotheses. Our proofs utilize knot filtered embedded contact homology, which was first introduced and computed by Hutchings for the standard transverse unknot in the irrational ellipsoids and further developed in our previous work. We continue our development of nontoric methods for embedded contact homology and establish the knot filtration on the embedded contact homology chain complex of the standard tight three-sphere with respect to positive $T(p,q)$ torus knots, where there are nonvanishing differentials. We also obtain obstructions to the existence of exact symplectic cobordisms between positive transverse torus knots.