  Given a transverse knot $K$ in a three dimensional contact manifold
$(Y,\alpha)$, in [13] Colin, Ghiggini, Honda and Hutchings define a hat version
of embedded contact homology for $K$, that we call $\widehat{ECK}(K,Y,\alpha)$,
and conjecture that it is isomorphic to the knot Floer homology
$\widehat{HFK}(K,Y)$. We define here a full version $ECK(K,Y,\alpha)$ and
generalise the definitions to the case of links. We prove then that, if $Y =
S^3$, $ECK$ and $\widehat{ECK}$ categorify the (multivariable) Alexander
polynomial of knots and links, obtaining expressions analogue to that for knot
and link Floer homologies in the plus and, respectively, hat versions.
