A geometric classification of rod complements in the 3-torus

TL;DR

This paper provides a complete topological classification of rod complements in the 3-torus, advancing the understanding of their geometric structures in crystallography.

Contribution

It offers a comprehensive topological classification of all rod complements in the 3-torus, extending previous partial geometric classifications.

Findings

Complete classification of rod complements in the 3-torus

Topological methods applied to geometric classification

Clarification of geometric structures in crystallography

Abstract

Rod packings are used in crystallography to describe crystal structures with linear or zigzag chains of particles, and each rod packing can be topologically viewed as a collection of disjoint geodesics in the 3-torus. Hui and Purcell developed a method to study the complements of rods in the 3-torus with the use of 3-dimensional geometry and tools from the 3-sphere, and they partially classified the geometry of some families of rod complements in the 3-torus. In this paper, we provide a complete classification of the geometry of all rod complements in the 3-torus using topological arguments.