AI 中文总结
研究非环面拉格朗日子流形辛（不）打结性，构造受奥克利 - 厄舍拉格朗日量启发，比较其与相关对象关系；还研究\(Q^3\)中单调拉格朗日环面，计算相关不变量，证明不可位移性并联系镜像对称。
AI 中文摘要
本文的第一个主要主题是研究非环面拉格朗日子流形的辛（不）打结性。我们对这类拉格朗日量的构造受奥克利 - 厄舍广义波尔特罗维奇拉格朗日量启发，依赖于对辛约化方法的改编，可追溯到切卡诺夫 - 施伦克以及麦克杜夫的探针在非阿贝尔群作用情形。我们比较奥克利 - 厄舍拉格朗日量与其（不）打结的同类，发现它们在不同情形下：（1）非微分同胚；（2）微分同胚但非拉格朗日同痕；（3）拉格朗日同痕但非哈密顿同痕；（4）哈密顿同痕。当紧致化环境空间时，它们之间的这种关系经常变化。第二个主要主题是深入研究通过将我们的构造应用于三维二次曲面\(Q^3\)得到的单调拉格朗日环面。计算谢卢欣 - 托诺诺格 - 维亚纳的\(\Psi\)不变量的通用形变，我们找到一个在强意义下打结的环面：它与二维二次曲面中任何维亚纳环面的比兰提升不是哈密顿同痕的。此外，我们研究\(Q^3\)中奥克利 - 厄舍环面的枚举性质并证明其不可位移性，解决了奥克利 - 厄舍提出的一个开放问题。我们通过证明\(Q^3\)的单调富卡亚范畴的分裂生成结果将此与镜像对称联系起来，该结果受阿博扎伊德 - 迪奥戈关于球面余切丛的类似结果启发并可与之比较。
英文摘要
The first main theme of this paper is to study symplectic (un-)knottedness of Lagrangian submanifolds which are not tori. Our construction of such Lagrangians is inspired by the generalized Polterovich Lagrangians of Oakley--Usher and relies on an adaption of symplectic reduction methods going back to Chekanov--Schlenk and McDuff's probes to the case of non-abelian group actions. We compare Oakley--Usher Lagrangians to their (un-)knotted cousins to find that, depending on the situation, they are: (1) not diffeomorphic, (2) diffeomorphic, but not Lagrangian isotopic; (3) Lagrangian isotopic, but not Hamiltonian isotopic, (4) Hamiltonian isotopic. This relationship between them frequently changes when the ambient space is compactified. The second main theme is an in-depth study of monotone Lagrangian tori obtained from appyling our constructions to the three-dimensional quadric $Q^3$. Computing the versal deformation of the $Ψ$-invariant of Shelukhin--Tonkonog--Vianna, we find a torus which is knotted in a strong sense: it is not Hamiltonian isotopic to the Biran-lift of any Vianna torus in the two-dimensional quadric. Furthermore, we investigate enumerative properties of the Oakley--Usher torus in $Q^3$ and prove its non-displaceability, settling an open question asked by Oakley--Usher. We relate this to mirror symmetry by proving a split-generation result for the monotone Fukaya category of $Q^3$, which is inspired by, and can be compared to a similar result of Abouzaid--Diogo for cotangent bundles of the sphere.
Comments56 pages, 5 figures, several minor changes
