Construction and Classification of Combinatorial Weaving Diagrams

TL;DR

This paper presents a systematic algorithm for constructing and classifying periodic Euclidean weaving diagrams, which are a type of four-regular planar tiling with crossing information, using combinatorial methods and equivalence classes.

Contribution

It introduces a new algorithm for constructing weaving diagrams and a classification scheme based on crossing numbers, expanding understanding of their combinatorial structure.

Findings

Developed a systematic construction algorithm for weaving diagrams.

Defined equivalence classes using crossing-matrices.

Classified structures by minimum crossings per unit cell.

Abstract

This paper introduces a new systematic algorithm for constructing periodic Euclidean weaving diagrams with combinatorial arguments. It is shown that such a weaving diagram can be considered as a specific type of four-regular periodic planar tiling with over or under information at each vertex. Therefore, a weaving diagram can be constructed using two sets of cycles, one to build a tiling, and a second to define the crossing information. However, this construction method does not guarantee the uniqueness of the diagram, so we define the notion of equivalence classes of weaving diagrams using the concept of crossing-matrices. Finally, we present a classification of our periodic structures according to the minimum number of crossings on a unit cell.