Celoria Daniele, Mahler Barbara I
Mathematical Institute, University of Oxford, Radcliffe Observatory, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG, UK.
Proc Math Phys Eng Sci. 2022 May 25;478(2261):20210709. doi: 10.1098/rspa.2021.0709.
In this paper, we study how randomly generated knots occupy a volume of space using topological methods. To this end, we consider the evolution of the first homology of an immersed metric neighbourhood of a knot's embedding for growing radii. Specifically, we extract features from the persistent homology (PH) of the Vietoris-Rips complexes built from point clouds associated with knots. Statistical analysis of our data shows the existence of increasing correlations between geometric quantities associated with the embedding and PH-based features, as a function of the knots' lengths. We further study the variation of these correlations for different knot types. Finally, this framework also allows us to define a simple notion of deviation from ideal configurations of knots.
在本文中，我们使用拓扑方法研究随机生成的纽结如何占据空间体积。为此，我们考虑随着半径增加，纽结嵌入的浸入度量邻域的第一同调群的演化。具体而言，我们从与纽结相关的点云构建的Vietoris-Rips复形的持久同调（PH）中提取特征。我们数据的统计分析表明，与嵌入相关的几何量和基于PH的特征之间存在越来越强的相关性，这是纽结长度的函数。我们进一步研究了不同纽结类型下这些相关性的变化。最后，这个框架还使我们能够定义一个与纽结理想构型偏差的简单概念。
