Reeb dynamics of the link of the $A_n$ singularity

Abstract

The link of the $A_n$ singularity, $L_{A_n}\subset {ℂ}^3$ 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 $L\left(n+1,n\right)$, equipped with its canonical contact structure ${\xi }_{std}$.