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Updated: Dec 10, 2025

Self-assembling Morphologies Obtained from Helical Polycarbodiimide Copolymers and Their Triazole Derivatives
Published on: February 7, 2017
Isogonal weavings on the sphere: knots, links, polycatenanes
Michael O'Keeffe1, Michael M J Treacy2
1School of Molecular Sciences, Arizona State University, Tempe, Arizona 85287, USA.
Abstract:
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.
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