Classification of doubly periodic untwisted (p,q)-weaves by their crossing number

TL;DR

This paper classifies a specific type of doubly periodic weaves, called untwisted (p,q)-weaves, based on their minimal crossing number, and introduces a crossing matrix to define equivalence classes.

Contribution

It provides a novel classification method for untwisted (p,q)-weaves using crossing numbers and introduces a crossing matrix for equivalence class determination.

Findings

Classification of untwisted (p,q)-weaves by crossing number

Introduction of crossing matrix for equivalence classes

Representation of weave diagrams as quadrivalent periodic planar graphs

Abstract

A weave is the lift to the Euclidean thickened plane of a set of infinitely many planar crossed geodesics, that can be characterized by a number of sets of threads describing the organization of the non-intersecting curves, together with a set of crossing sequences representing the entanglements. In this paper, the classification of a specific class of doubly periodic weaves, called untwisted (p,q)-weaves, is done by their crossing number, which is the minimum number of crossings that can possibly be found in a unit cell of its infinite weaving diagrams. Such a diagram can be considered as a particular type of quadrivalent periodic planar graph with an over or under information at each vertex, whose unit cell corresponds to a link diagram in a thickened torus. Moreover, considering that a weave is not uniquely defined by its sets of threads and its crossing sequences, we also specify the notion of equivalence classes by introducing a new parameter, called crossing matrix.