  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) 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 fibered spaces. We provide Morse-Bott methods, using a doubly
filtered complex and the energy filtered perturbed Seiberg-Witten theory
developed by Hutchings and Taubes, and use them to compute the T(2,q) knot
filtered embedded contact homology, for q odd and positive. In the sequel we
complete the computation for positive T(p,q) knots (where there is a
nonvanishing differential) and use our results to deduce quantitative existence
results for torus knotted Reeb dynamics on the tight 3-sphere and the mean
action of area preserving diffeomorphisms of once punctured surfaces of
arbitrary genus arising as Seifert surfaces of positive torus knots.
