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 $\left(L_{A_n}, \xi_0 \right)$\mathrm{.}Inaddition,weprovethat$\left(L_{A_n}, \xi_0 \right)$ is contactomorphic to the Lens space $L(n+1,n)$, equipped with its canonical contact structure ${\xi }_{std}$.