Combinatorial Reeb dynamics on punctured contact 3-manifolds

TL;DR

This paper explores the combinatorial structure of Reeb dynamics on punctured contact 3-manifolds, linking Reeb orbits to words of Reeb chords, and introduces algebraic tools to analyze holomorphic curves, leading to new examples of tight contact manifolds with vanishing contact homology.

Contribution

It develops a combinatorial framework connecting Reeb orbits to Reeb chords and introduces algebraic methods for studying holomorphic curves in contact surgery cobordisms.

Findings

Reeb orbits correspond to cyclic words of Reeb chords.

Homology classes and Conley-Zehnder indices are computed diagrammatically.

Constructs examples of tight contact manifolds with vanishing contact homology.

Abstract

Let $\Lambda^{\pm} = \Lambda^{+} \cup \Lambda^{-} \subset (\mathbb{R}^{3}, \xi_{std})$ be a contact surgery diagram determining a closed, connected contact $3$-manifold $(S^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$ and an open contact manifold $(\mathbb{R}^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$. Following arXiv:0911.0026 and arXiv:1906.07228 we demonstrate how $\Lambda^{\pm}$ determines a family $\alpha_{\epsilon}$ of contact forms on $(\mathbb{R}^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$ whose closed Reeb orbits are in one-to-one correspondence with cyclic words of composable Reeb chords on $\Lambda^{\pm}$. We compute the homology classes and integral Conley-Zehnder indices of these orbits diagrammatically and develop algebraic tools for studying holomorphic curves in surgery cobordisms between the $(\mathbb{R}^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$. These new techniques are used to describe the first known examples of closed, tight contact manifolds with vanishing contact homology. They are contact $\frac{1}{k}$ surgeries along the right-handed, $tb=1$ trefoil for $k > 0$, which are known to have non-zero Heegaard-Floer contact classes by arXiv:math/0404135.