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.
Original language | English (US) |
---|---|
Pages (from-to) | 498-506 |
Number of pages | 9 |
Journal | Acta Crystallographica Section A: Foundations of Crystallography |
Volume | 78 |
DOIs | |
State | Published - Nov 1 2022 |
Keywords
- catenanes
- piecewise-linear embeddings
- tangled structures
- theta graph
ASJC Scopus subject areas
- Structural Biology
- Biochemistry
- General Materials Science
- Condensed Matter Physics
- Physical and Theoretical Chemistry
- Inorganic Chemistry