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- Title
Piecewise‐linear embeddings of decussate extended θ graphs and tetrahedra.
- Authors
O'Keeffe, Michael; Treacy, Michael M. J.
- Abstract
An nθ graph is an n‐valent graph with two vertices. From symmetry considerations, it has vertex–edge transitivity 1 1. Here, they are considered extended with divalent vertices added to the edges to explore the simplest piecewise‐linear tangled embeddings with straight, non‐intersecting edges (sticks). The simplest tangles found are those with 3n sticks, transitivity 2 2, and with 2⌊(n − 1)/2⌋ ambient‐anisotopic tangles. The simplest finite and 1‐, 2‐ and 3‐periodic decussate structures (links and tangles) are described. These include finite cubic and icosahedral and 1‐ and 3‐periodic links, all with minimal transitivity. The paper also presents the simplest tangles of extended tetrahedra and their linkages to form periodic polycatenanes. A vertex‐ and edge‐transitive embedding of a tangled srs net with tangled and polycatenated θ graphs and vertex‐transitive tangled diamond (dia) nets are described.
- Subjects
SYMMETRY; FINITE, The; TETRAHEDRA; KNOT theory; DIAMONDS; CATENANES
- Publication
Acta Crystallographica. Section A, Foundations & Advances, 2022, Vol 78, Issue 6, p498
- ISSN
2053-2733
- Publication type
Article
- DOI
10.1107/S2053273322008750