论文标题
刺穿接触的组合REEB动力学3- manifolds
Combinatorial Reeb dynamics on punctured contact 3-manifolds
论文作者
论文摘要
令$λ^{\ pm} =λ^{+} \cupλ^{ - } \ subset（\ mathbb {\ mathbb {r}^{3}，ξ_{std}）$ be contact手术the contact手术图确定一个封闭的，连接的，连接的$ 3 $ 3 $ -MANIFOLD $（MANIFOLD $（s^pmmanifold $） ξ_{λ^{\ pm}}）$和开放式触点歧管$（\ Mathbb {r}^{3} _ {λ^{\ pm}}，ξ_{λ^{\ pm}}）$。以下arXiv：0911.0026和arxiv：1906.07228我们演示了$λ^{\ pm} $如何确定$（\ mathbb {r}^{r}^{3} _ {3} _ {λ^λ^{λ^{\ pm}}的家庭$α_ε$的联系表格，与$λ^{\ pm} $上的可合曲reeb和弦的循环单词一对一。 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 $(\mathbb{R}^{3}_{Λ^{\pm}}, ξ_{Λ^{\pm}})$. 这些新技术用于描述具有消失的接触同源性的封闭，紧密接触歧管的第一个已知示例。他们是在右撇子上与$ \ frac {1} {k} $手术，$ tb = 1 $ trefoil for $ k> 0 $，众所周知，它们具有非零的Heegaard-loer Contact类，由Arxiv：Math/0404135。
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.
