We exhibit a distinctly low-dimensional dynamical obstruction to the existence of Liouville cobordisms: for any contact 3-manifold admitting an exact symplectic cobordism to the tight 3-sphere, every nondegenerate contact form admits an embedded Reeb orbit that is unknotted and has self-linking number -1. The same is true moreover for any contact structure on a closed 3-manifold that is reducible. Our results generalize an earlier theorem of Hofer-Wysocki-Zehnder for the 3-sphere, but use somewhat newer techniques: the main idea is to exploit the intersection theory of punctured holomorphic curves in order to understand the compactification of the space of so-called "nicely embedded" curves in symplectic cobordisms. In the process, we prove a local adjunction formula for holomorphic annuli breaking along a Reeb orbit, which may be of independent interest.

中文翻译：

---

未打结的 Reeb 轨道和很好嵌入的全纯曲线

我们对 Liouville cobordisms 的存在表现出明显的低维动力学障碍：对于任何接触 3-流形承认紧密的 3 球体的精确辛cobordism，每个非退化接触形式都承认一个嵌入的 Reeb 轨道，该轨道是未打结的并且具有自链接号-1。此外，对于可还原的闭合 3 歧管上的任何接触结构也是如此。我们的结果概括了 3 球体的 Hofer-Wysocki-Zehnder 的早期定理，但使用了一些较新的技术：主要思想是利用穿孔全纯曲线的相交理论来理解所谓的空间的紧化“很好地嵌入”辛cobordisms 曲线。在此过程中，我们证明了全纯环沿 Reeb 轨道断裂的局部附加公式，