A categorification of the Alexander polynomial in embedded contact homology

Given a transverse knot K in a three-dimensional contact manifold (Y, alpha) , Colin, Ghiggini, Honda and Hutchings defined a hat version (ECK) over cap (K, Y, alpha) of embedded contact homology for K and conjectured that it is isomorphic to the knot Floer homology (HFK) over cap (K, Y). We define here a full version ECK (K, Y, alpha) and generalize the definitions to the case of links. We prove then that if Y = S-3, then ECK and (ECK) over cap categorify the (multivariable) Alexander polynomial of knots and links, obtaining expressions analogous to that for knot and link Floer homologies in the minus and, respectively, hat versions.