Construction and Classification of Weaves

本研究的目的是将周期性组织定义、构造和分类为新的拓扑对象。为了构造编织，我们用组合论元改进了我们的方法，并开发了新的编织不变量来使用纽结理论对它们进行分类。特别是，我们开发了“多边形链接法”，它允许人们使用平铺理论来构建和分类组织。更具体地说，在给定平面的双周期瓦片和多边形链方法的情况下，通过指导如何用线束覆盖瓦片的边和顶点，我们引入了一种系统的算法来预测、识别、构造和分类编织图案和其他纠缠结构，如多环戊二烯和混合图案。然后，我们开始根据组织的对称群对其进行分类。

The purpose of this research was to define, construct, and classify periodic weaves as new topological objects. To construct weaves, our approach has been improved using combinatorial arguments and we developed new weaving invariants to classify them using knot theory. In particular, we developed the 'polygonal link methods' which allow one to build and classify weaves using tiling theory. More specifically, given a doubly periodic tiling of the plane and a polygonal link method, by instructing how to cover edges and vertices of the tiling by strands, we introduced a systematic algorithm to predict, distinguish, construct and classify weaving motifs and other entangled structures such as polycatenane and mixed motifs. Then, we started a classification of weaves by their symmetry groups.