Reeb Dynamics of the Link of the $A_n$ Singularity

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 $\xi_{std}$.