From knot filtered ECH to surface dynamics

Jo Nelson, Rice University

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Passcode: 998749

Contact topology is the study of maximally nonintegrable hyperplane fields on odd dimensional smooth manifolds. The contact one form uniquely determines a Hamiltonian-like vector field called the Reeb vector field on the manifold. I will explain some background on embedded contact homology (ECH), a Floer type gauge theory generated by closed Reeb currents.   I will discuss work in progress with Morgan Weiler on knot filtered ECH of open book decompositions of S^3 along T(2,q) torus knots.  This spectral invariant provides information about the dynamics of symplectomorphisms of the genus (q-1)/2 pages which are freely isotopic to rotation by 1/(2q) along the boundary. I will explain the interplay between the topology of the open book, its presentation as an orbi-bundle, and our computation of the knot filtered ECH chain complex.  I will describe how knot filtered ECH realizes the relationship between the action and linking of Reeb orbits and its application to the study of the Calabi invariant and periodic orbits of symplectomorphisms of the pages.