Cylindrical Contact Homology on Complements of Reeb Orbits

Let $V$ be a closed 3-manifold with a contact form $\lambda$, and let $L$ be a link consisting of closed orbits for the Reeb vector field of $\lambda$. We study the problem of defining cylindrical contact homology on the non-compact manifold $V \backslash L$, giving sufficient conditions on a class of forms so that the chain complex can be defined in the expected way. Under further technical assumptions we show that the homology of the complex up to isomorphism is independent of choices except possibly the values of certain Conley-Zehnder indices associated with $L$. A simple example shows that the homology can depend on the Conley-Zehnder indices of the components of $L$.