带洞切触3-流形上的组合Reeb动力学
Combinatorial Reeb dynamics on punctured contact 3-manifolds
AI总结：
该研究利用切触手术图构造带洞切触3-流形上的切触形式，建立闭Reeb轨道与Reeb弦循环字的对应，发展全纯曲线研究工具，并构造出首批切触同调消失但Heegaard-Floer切触类非零的紧闭切触流形例子。
摘要：
设$\Lambda^{\pm} = \Lambda^{+} \cup \Lambda^{-} \subset (\mathbb{R}^{3}, \xi_{std})$为切触手术图，它确定了一个闭的连通切触3-流形$(S^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$以及一个开切触流形$(\mathbb{R}^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$。参考arXiv:0911.0026与arXiv:1906.07228的工作，我们证明了$\Lambda^{\pm}$如何确定$(\mathbb{R}^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$上的一族切触形式$\alpha_\varepsilon$，其闭Reeb轨道与$\Lambda^{\pm}$上可复合Reeb弦的循环字一一对应。我们通过图论方法计算了这些轨道的同调类与整值Conley-Zehnder指标，并开发了代数工具，用于研究$(\mathbb{R}^{3}_{\Lambda^{\pm}}, \xi_{\Lambda^{\pm}})$之间手术配边中的全纯曲线。\n这些新技术被用于描述首批已知的切触同调消失的紧闭切触流形例子。它们是沿右手、$tb=1$的三叶结进行的切触$\frac{1}{k}$手术（$k > 0$），根据arXiv:math/0404135的结果，这类流形具有非零的Heegaard-Floer切触类。
英文摘要：
Let $Λ^{\pm} = Λ^{+} \cup Λ^{-} \subset (\mathbb{R}^{3}, ξ_{std})$ be a contact surgery diagram determining a closed, connected contact $3$-manifold $(S^{3}_{Λ^{\pm}}, ξ_{Λ^{\pm}})$ and an open contact manifold $(\mathbb{R}^{3}_{Λ^{\pm}}, ξ_{Λ^{\pm}})$. Following arXiv:0911.0026 and arXiv:1906.07228 we demonstrate how $Λ^{\pm}$ determines a family $α_ε$ of contact forms on $(\mathbb{R}^{3}_{Λ^{\pm}}, ξ_{Λ^{\pm}})$ whose closed Reeb orbits are in one-to-one correspondence with cyclic words of composable Reeb chords on $Λ^{\pm}$. 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 $(\mathbb{R}^{3}_{Λ^{\pm}}, ξ_{Λ^{\pm}})$. 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.
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