四维流形中光滑纽结曲面的拓扑平凡性

等价意义下的稳定奇异纽结曲面

Stably exotic knotted surfaces up to equivalence

该研究证明了4维流形中存在一类稳定奇异纽结曲面，其拓扑等价但经任意次内部、外部稳定化后仍不光滑等价，丰富了纽结曲面的等价性理论。

Exotically knotted disks and complex curves

This paper studies properly embedded surfaces in the 4-ball that are exotically knotted (i.e., topologically but not smoothly isotopic), and leverages this local phenomenon to study surfaces in larger 4-manifolds. The main results provide a new construction of exotically knotted surfaces, including exotic slice surfaces of all genera in the 4-ball and exotic closed surfaces in larger 4-manifolds. The construction is well-suited to the complex and symplectic settings, providing the first examples of exotically knotted complex curves and symplectic 2-spheres. Along the way, we articulate some di…

exotic 结合的 2 球面及其补集的基本群

Exotically knotted 2-spheres and the fundamental groups of their complements

该研究展示了一种构造包含特定 2 链的 4 维 manifold 的方法，其 2 链群可为任意有限生成群，并证明这些链的组成部分具有空同源性质。

Bridge trisections of knotted surfaces in 4--manifolds

We prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a \emph{generalized bridge trisection}, extends the authors' definition of bridge trisections for surfaces in $S^4$. Using this new construction, we give diagrammatic representations called \emph{shadow diagrams} for knotted surfaces in 4--manifolds. We also provide a low-complexity classification f…

A Levine-Tristram invariant for knotted tori

Echeverria recently introduced an invariant for a smoothly embedded torus in a homology $S^1\times S^3$, using gauge theory for singular connections. We define a new topological invariant of such an embedded torus, analogous to the classical Levine-Tristram invariant of a knot. In the 3-dimensional situation, a count of singular connections on a knot complement reproduces the Levine-Tristram invariant. We compute the invariant for a number of examples embedded tori, and show that our topological invariant is the same as what one might expect from Echeverria's invariant. Langte Ma has subsequen…

Knotted surfaces in 4-manifolds by Knot surgery and Stabilization

Given a simply-connected closed 4-manifold $X$ and a smoothly embedded oriented surface $Σ$, various constructions based on Fintushel-Stern knot surgery have produced new surfaces in $X$ that are pairwise homeomorphic to $Σ$, but not diffeomorphic. We prove that for all known examples of surface knots constructed from knot surgery operations that preserve the fundamental group of the complement of surface knots, they become pairwise diffeomorphic after stabilizing by connected summing with one $S^2\tilde{\times}S^2$. When $X$ is spin, we show in addition that any surfaces obtained by a knot su…

四维稳定合痕

Stable isotopy in four dimensions

我们在许多单连通4-流形中构造了拓扑同位素但光滑不同的纽结球的无限族，这些纽结球在通过与$S^2 \times S^2$的连通求和稳定化后变得光滑同位素，并且因此，对于这样的4-流形，构造了类似的正数量曲率的反同态和度量族。我们还构建家庭的顺利不同的链接，其相应的适当的子链接是顺利的同位素，成为稳定后顺利同位素。

二重点手术与曲面构形

Double point surgery and configurations of surfaces

本文在四维流形的浸入曲面上引入了一种新的操作--二重点操作，并利用它构造了许多四维流形中曲面的纽结构形。以分支覆盖为例，我们在具有复杂不动点集的单连通4-流形上产生了Z/mxZ/n的光滑奇异作用。

四维流形中的纽结曲面

Knotted surfaces in 4-manifolds

Fintushel和Stern证明了：如果S \subset X是辛4-流形中的辛曲面，使得S具有单连通补和非负自交，则存在无穷多个拓扑等价但光滑不同的嵌入曲面与S同源.在这里，我们将这个结果推广到包括辛曲面，其自相交的下界为2-2g，其中g是S的亏格。我们使用的工具，从Heegaard Floer理论，并包括几个结果，可能是独立的利益。具体来说，我们给出了一个类似的Ozsvath-Szabo不变量的Fintushel-Stern结手术公式的Seiberg-Witten不变量，无论是封闭的4-流形和流形的边界。这是基于一个公式的Ozsvath-Szabo不变量的结果的对数变换，类似于一个获得的摩根-Mrowka-Szabó的塞伯格-威滕不变量，和结果的Ozsvath-Szabo不变量的纤维总和由于作者和Jabuka。此外，我们还计算了Riemann曲面上“大”度圆丛的扭曲Heegaard弗洛尔同调.

位错与纤维化：扭结近晶缺陷的拓扑结构

Dislocations and Fibrations: The Topological Structure of Knotted Smectic Defects

本研究探究近晶液晶中扭结缺陷的拓扑性质，分析螺位错、刃型位错等缺陷与扭结纤维化的关联，给出点缺陷数量下界，深化了与焦锥畴的类比，为相关拓扑理论提供研究平台。

Symplectic surfaces and bridge position

We give a new characterization of symplectic surfaces in CP^2 via bridge trisections. Specifically, a minimal genus surface in CP^2 is smoothly isotopic to a symplectic surface if and only if it is smoothly isotopic to a surface in transverse bridge position. We discuss several potential applications, including the classification of unit 2-knots, establishing the triviality of Gluck twists, the symplectic isotopy problem, Auroux's proof that every symplectic 4-manifold is a branched cover over CP^2, and the existence of Weinstein trisections. The proof exploits a well-known connection between …

四维流形中曲面的扭曲旋转变形

Modifying surfaces in 4-manifolds by twist spinning

在本文中，给定一个纽结K，对于任意整数m，我们通过对Sigma进行手术，从光滑4-流形X中的光滑嵌入曲面Sigma构造了一个新的曲面Sigma_K（m）。这种手术是在Fintushel和Stern提出的“边缘手术”的基础上进行修改，通过进行额外的加捻纺纱。我们研究了手术后（X，Sigma）的同胚型和自同构型。主要结果之一是，对于一定的对（X，Sigma），Sigma_K（m）的光滑型可以很容易地由纽结K的Alexander多项式来区分，而同胚型则取决于纽结和扭数.特别是，我们在CP ^[2]中得到了纽结曲面的新例子，它们与复曲线不是同素的，而是拓扑非纽结的。

4维流形中曲面的奇异纽结与对称性

Exotic knottings and symmetries of surfaces in 4-manifolds

该研究通过环境对称性，结合rim surgery、凸几何与双曲几何，构造4维流形中的射影刚性、拓扑灵活曲面及全测地曲面，揭示奇异纽结与精细对称性现象。