Piecewise‐linear embeddings of decussate extended θ graphs and tetrahedra. Issue 6 (21st October 2022)

• Record Type: Journal Article
• Authors: O'Keeffe, Michael; Treacy, Michael M. J.
• Is Part Of: Acta crystallographica. Volume 78:Issue 6(2022)
• Page Start: 498
• Page End: 506
• Publication Date: 2022-10-21
• Subjects: theta graph -- piecewise‐linear embeddings -- tangled structures -- catenanes; Crystallography -- Periodicals; Condensed matter -- Periodicals; 548
• DOI: 10.1107/S2053273322008750
• Languages: English
• ISSNs: 2053-2733

Abstract

Abstract : The simplest (lowest transitivity) piecewise‐linear tangles of extended n θ graphs, including 1‐, 2‐ and 3‐periodic linked tangles, are presented. Abstract : An n θ 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 3 n 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.