Dynamics of Seifert Surfaces of Torus Knots Via ECH

Embedded contact homology (ECH) is a diffeomorphism invariant of three-manifolds due to Hutchings, defined using a contact form. This very diffeomorphism invariance makes it quite useful when studying contact dynamics, because it is possible to apply calculations using simpler contact forms to situations involving more complex ones. We will outline how a knot filtration on ECH was used in a 2015 paper of Hutchings to identify low mean action periodic orbits of disk maps, as well as several more recent generalizations due to other authors. We will then explain the correspondence between the existence of an action function and the construction of a contact three-manifold (via a mapping torus) when starting with a specific surface symplectomorphism. Finally, we will mention how the ECH computations change by analyzing the case of T(2,3) and its genus one Seifert surface. Based on work in progress with Jo Nelson. Note: The material discussed in this talk will differ from that of Jo Nelson's February 27 talk. However, enough background will be given to make this talk self-contained.