论文标题
歧管中的随机着色
Random Colorings in Manifolds
论文作者
论文摘要
我们开发了一种在任意维度中构建随机流形和子曼群的通用方法。该方法基于将颜色与三角歧管的顶点相关联，如Sheffield和Yadin（2014）的三维空间曲线的最新工作中。我们根据Stiefel-Whitney类和其他属性来确定可以在哪些子手机上出现的条件。然后，我们考虑由随机着色顶点引起的随机子术。由于该模型生成子曼膜，因此它允许研究属性并使用在产生一般随机子复合物的过程中无法使用的工具。三角三角球中3种颜色的情况产生了随机结和链接。在这种情况下，我们回答了De Crouy-Chanel和Simon（2019）提出的一个问题，表明产生无结的可能性是指数型衰减的。在d维歧管中k颜色的一般情况下，我们研究了不同编码的随机亚策略，因为三角剖分中的顶点数量增加。我们计算预期的欧拉特征，并讨论与同源渗透和其他拓扑特性的关系。最后，我们探讨了一种通过生成随机子手势来搜索拓扑问题解决方案的方法。我们描述了在4维球中搜索低生物表面的计算机实验，其边界是三维球中给定的结。
We develop a general method for constructing random manifolds and submanifolds in arbitrary dimensions. The method is based on associating colors to the vertices of a triangulated manifold, as in recent work for curves in 3-dimensional space by Sheffield and Yadin (2014). We determine conditions on which submanifolds can arise, in terms of Stiefel-Whitney classes and other properties. We then consider the random submanifolds that arise from randomly coloring the vertices. Since this model generates submanifolds, it allows for studying properties and using tools that are not available in processes that produce general random subcomplexes. The case of 3 colors in a triangulated 3-ball gives rise to random knots and links. In this setting, we answer a question raised by de Crouy-Chanel and Simon (2019), showing that the probability of generating an unknot decays exponentially. In the general case of k colors in d-dimensional manifolds, we investigate the random submanifolds of different codimensions, as the number of vertices in the triangulation grows. We compute the expected Euler characteristic, and discuss relations to homological percolation and other topological properties. Finally, we explore a method to search for solutions to topological problems by generating random submanifolds. We describe computer experiments that search for a low-genus surface in the 4-dimensional ball whose boundary is a given knot in the 3-dimensional sphere.
