  Embedded contact knot homology (ECK) is a variation on Embedded contact
homology (ECH), defined with respect to an open book decomposition compatible
with a contact structure on some 3-manifold, M. The knot in question is given
by the (null-homologous) binding of the open book and the chain complex is
defined in terms of closed orbits of the Reeb vector field and certain
pseudoholomorphic curves in the symplectization of the knot complement. In this
thesis we first generalize this construction to the case of rational open book
decompositions, allowing us to define ECK for rationally null-homologous knots.
In its most general form this is a bi-filtered chain complex whose homology
yields ECH of the closed manifold. There is also a hat version of ECK in this
situation which is equipped with an Alexander grading equivalent to that in the
Heegaard Floer setting, categorifies the Alexander polynomial, and is
conjecturally isomorphic to the hat version of knot Floer homology, HFK. The
main result of this thesis is a large negative n-surgery formula for ECK.
Namely, we start with an (integral) open book decomposition of a manifold with
binding K and compute, for large n, ECK of the knot K(-n) obtained by
performing (-n)-surgery on K. This formula agrees with Hedden's large n-surgery
formula for HFK, providing supporting evidence towards the conjectured
equivalence between the two theories. Along the way, we also prove that ECK is,
in many cases, independent of the choices made to define it, namely the almost
complex structure on the symplectization and the homotopy type of the contact
form. We also prove that, in the case of integral open book decompositions, the
hat version of ECK is supported in Alexander gradings less than or equal to
twice the genus of the knot.
