论文标题
外部打结的磁盘和复杂曲线
Exotically knotted disks and complex curves
论文作者
论文摘要
本文研究了在四球中正确嵌入的表面，这些表面是在四球上进行的，这些表面是在外向打结的（即拓扑，但不是平滑的同位素），并利用这种局部现象来研究较大的4个序列中的表面。主要结果提供了外向打结的表面的新结构，包括四球和外来封闭式较大的4个manifolds中所有属的外来切片表面。该结构非常适合复杂而符合性的设置，提供了外向打结的复杂曲线和符号曲线的第一个示例。在此过程中，我们阐明了一些图形工具，用于构建符号表面和复杂曲线。 我们还使用局部打结来研究打结组的地理问题，构建了封闭的外向打结表面的第一个例子，简单地连接了4个模型，其结组包含非阿比亚自由亚组，因此，预计在外科理论意义上不会是“好”群体。
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 diagrammatic tools for constructing symplectic surfaces and complex curves. We also use local knotting to investigate the geography problem for knot groups, constructing the first examples of exotically knotted surfaces in closed, simply connected 4-manifolds whose knot groups contain nonabelian free subgroups, hence are not expected to be "good" groups in the sense of surgery theory.
