Georgia Institute of Technology College of Sciences

In 2016, Hutchings introduced a knot filtration on the embedded contact homology (ECH) chain complex in order to estimate the linking of periodic orbits of the Reeb vector field, with an eye towards applications to dynamics on the disk. Since then, the knot filtration has been computed for certain lens spaces by myself, and the "action-linking" relationship has been studied for generic contact forms on general three-manifolds by Bechara Senior-Hryniewicz-Salomao. In joint work with Jo Nelson, we study dynamics on surfaces with one boundary component by computing the knot filtration on the ECH chain complex of positive torus knots in S^3. This requires us to understand the contact form as both a prequantization orbibundle and as a periodic open book with positive fractional Dehn twist coefficient. We will focus on the latter point of view to describe how the computation works and the prospects for extending it to more general open books.