#### Vol. 10, No. 3, 2017

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Reeb dynamics of the link of the $A_n$ singularity

### Leonardo Abbrescia, I. Huq-Kuruvilla, J. Nelson and N. Sultani

Vol. 10 (2017), No. 3, 417–442
##### 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 ${\xi }_{std}$.

##### Keywords
contact geometry, contact topology, Conley–Zehnder index, $A_n$ singularity, Reeb dynamic, Maslov index
##### Mathematical Subject Classification 2010
Primary: 37B30, 53D35, 57R17
Secondary: 53D42