  The link of the $A_n$ singularity, $L_{A_n} \subset \mathbb{C}^3$ admits a
natural contact structure $\xi_0$ coming from the set of complex tangencies.
The canonical contact form $\alpha_0$ associated to $\xi_0$ is degenerate and
thus has no isolated Reeb orbits. We show that there is a nondegenerate contact
form for a contact structure equivalent to $\xi_0$ that has two isolated simple
periodic Reeb orbits. We compute the Conley-Zehnder index of these simple
orbits and their iterates. From these calculations we compute the positive
$S^1$-equivariant symplectic homology groups for $\left(L_{A_n}, \xi_0
\right)$. In addition, we prove that $\left(L_{A_n}, \xi_0 \right)$ is
contactomorphic to the Lens space $L(n+1,n)$, equipped with its canonical
contact structure $\xi_{std}$.

Symplectic Geometry