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Isogonal weavings on the sphere: knots, links, polycatenanes
Author(s) -
O'Keeffe Michael,
Treacy Michael M. J.
Publication year - 2020
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.742
H-Index - 83
ISSN - 2053-2733
DOI - 10.1107/s2053273320010669
Subject(s) - icosahedral symmetry , transitive relation , vertex (graph theory) , torus , combinatorics , mathematics , piecewise linear function , pure mathematics , geometry , graph
Mathematical knots and links are described as piecewise linear – straight, non‐intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry (`vertex'‐transitive). Corner‐ and stick‐transitive structures are termed regular. No regular knots are found. Regular links are cubic or icosahedral and a complete account of these (36 in number) is given, including optimal (thickest‐stick) embeddings. Stick 2‐transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. The relevance of this work to materials chemistry and biochemistry is noted.