## 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) |
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Pages (from-to) | 498-506 |

Number of pages | 9 |

Journal | Acta Crystallographica Section A: Foundations and Advances |

Volume | 78 |

DOIs | |

State | Published - Nov 1 2022 |

Externally published | Yes |

## Keywords

- catenanes
- piecewise-linear embeddings
- tangled structures
- theta graph

## ASJC Scopus subject areas

- Structural Biology
- Biochemistry
- Materials Science(all)
- Condensed Matter Physics
- Physical and Theoretical Chemistry
- Inorganic Chemistry