Piecewise-linear embeddings of decussate extended θ graphs and tetrahedra

Michael O'Keeffe, Michael M.J. Treacy

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)498-506
Number of pages9
JournalActa Crystallographica Section A: Foundations of Crystallography
StatePublished - Nov 1 2022


  • 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


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