Combinatorial Reeb dynamics on punctured contact 3–manifolds

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 standard-at-infinity 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 using a simultaneous framing of all orbits naturally determined by the surgery diagram, providing a (typically non-canonical) $\mathbb{Z}$-grading on the chain complexes underlying the "hat" version of contact homology as defined in arXiv:1004.2942. Using holomorphic foliations, algebraic tools for studying holomorphic curves in symplectizations of and surgery cobordisms between the $(\mathbb{R}^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$ are developed. We use these computational tools to provide the first examples of closed, tight, contact manifolds with vanishing contact homology -- 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.