  Let $\Lambda^{\pm} = \Lambda^{+} \cup \Lambda^{-} \subset (\mathbb{R}^{3},
\xi_{std})$ be a contact surgery diagram determining a closed, connected
contact $3$-manifold $(S^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$ and an open
contact manifold $(\mathbb{R}^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$.
Following arXiv:0911.0026 and arXiv:1906.07228 we demonstrate how
$\Lambda^{\pm}$ determines a family $\alpha_{\epsilon}$ of contact forms on
$(\mathbb{R}^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$ whose closed Reeb
orbits are in one-to-one correspondence with cyclic words of composable Reeb
chords on $\Lambda^{\pm}$. We compute the homology classes and integral
Conley-Zehnder indices of these orbits diagrammatically and develop algebraic
tools for studying holomorphic curves in surgery cobordisms between the
$(\mathbb{R}^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$.
  These new techniques are used to describe the first known examples of closed,
tight contact manifolds with vanishing contact homology. They are contact
$\frac{1}{k}$ surgeries along the right-handed, $tb=1$ trefoil for $k > 0$,
which are known to have non-zero Heegaard-Floer contact classes by
arXiv:math/0404135.
