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On the geometry of rod packings in the 3-torus

Connie On Yu Hui, Jessica S. Purcell

Abstract

Rod packings in the 3-torus encode information of some crystal structures in crystallography. They can be viewed as links in the 3-torus, and tools from 3-manifold geometry and topology can be used to study their complements. In this paper, we initiate the use of geometrisation to study such packings. We analyse the geometric structures of the complements of simple rod packings, and find families that are hyperbolic and Seifert fibred.

On the geometry of rod packings in the 3-torus

Abstract

Rod packings in the 3-torus encode information of some crystal structures in crystallography. They can be viewed as links in the 3-torus, and tools from 3-manifold geometry and topology can be used to study their complements. In this paper, we initiate the use of geometrisation to study such packings. We analyse the geometric structures of the complements of simple rod packings, and find families that are hyperbolic and Seifert fibred.
Paper Structure (12 sections, 22 theorems, 4 equations, 8 figures)

This paper contains 12 sections, 22 theorems, 4 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Organization
  3. Acknowledgements
  4. Preliminaries
  5. Decompositions into ideal octahedra
  6. Geometry of invariant cubic rod packings in crystallography
  7. One and two arbitrary rods in the $3$-torus
  8. The standard rods plus another rod
  9. Irreducibility and boundary-irreducibility
  10. Non-existence of essential tori
  11. Non-existence of essential annuli
  12. A family of hyperbolic links in the $3$-torus

Key Result

Lemma 2.2

Let $(a,b,c)\in \mathbb{Z}^3{\smallsetminus}\{(0,0,0)\}$. The map $R_e\colon [0,1]\to \mathbb{T}^3$ associated to the $(a,b,c)$-rod $R$ represents a simple closed curve in $\mathbb{T}^3$ if and only if $\mathrm{gcd}(a,b,c)=1$.

Figures (8)

  • Figure 1: $\mathbb{T}^3 {\smallsetminus} (R_x\cup R_y\cup R_z)$ and $\mathbb{S}^3 {\smallsetminus} (C_x\cup C_y\cup C_z)$.
  • Figure 2: Octahedral decomposition of $\mathbb{S}^3 {\smallsetminus} (C_x\cup C_y\cup C_z)$. Left: Link diagram with six edges and eight faces. Middle: Octahedron above link diagram (viewed from interior). Right: Octahedron below link diagram (viewed from exterior).
  • Figure 3: Octahedral decomposition of $\mathbb{T}^3 {\smallsetminus} (R_x\cup R_y\cup R_z)$. Left: $\mathbb{T}^3 {\smallsetminus} (R_x\cup R_y\cup R_z)$ with six edges and eight faces. Middle: The upper ideal octahedron (view from exterior). Right: The lower ideal octahedron (view from exterior)
  • Figure 4: Left: $\mathbb{T}^3 {\smallsetminus} (R_x\cup R_y\cup R_z)$ and $\mu_y$. Middle: Upper octahedron and half of $\mu_y$ (view from exterior). Right: Lower octahedron and second half of $\mu_y$ (view from exterior)
  • Figure 5: Identifying $h(\mu_y)$ in $\mathbb{S}^3 {\smallsetminus} (C_x\cup C_y\cup C_z)$. Left: Octahedron above link diagram and half of $\mu_y$ (view from interior). Middle: Octahedron below link diagram and half of $\mu_y$ (from exterior). Right: $\mathbb{S}^3 {\smallsetminus} (C_x\cup C_y\cup C_z)$ and $h(\mu_y)$
  • ...and 3 more figures

Theorems & Definitions (46)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Proposition 2.6
  • ...and 36 more