Torus knot filtered embedded contact homology of the tight contact 3-sphere

Knot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces L(n,n-1)$L(n,n-1)$ via a quotient by Weiler. While toric constructions can be used to understand the ECH chain complexes of many contact forms adapted to open books with binding the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3-sphere in terms of their presentation as open books and as Seifert fiber spaces. We provide Morse-Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg-Witten Floer theory developed by Hutchings and Taubes, and use them to compute the T(2,q)$T(2,q)$ knot filtered embedded contact homology, for q$q$ odd and positive.