Construction and Classification of Combinatorial Weaving Diagrams

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.