Combinatorial Reeb dynamics on punctured contact 3–manifolds

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