Isogonal weavings on the sphere: knots, links, polycatenanes. Issue 5 (21st August 2020)

• Record Type: Journal Article
• Authors: O'Keeffe, Michael; Treacy, Michael M. J.
• Is Part Of: Acta crystallographica. Volume 76:Issue 5(2020:Sep.)
• Page Start: 611
• Page End: 621
• Publication Date: 2020-08-21
• Subjects: weaves -- knots -- links -- catenanes; Crystallography -- Periodicals; Condensed matter -- Periodicals; 548
• DOI: 10.1107/S2053273320010669
• Languages: English
• ISSNs: 2053-2733

Abstract

Abstract : Non‐periodic knotted and linked structures that can be built using one kind of vertex (isogonal) and two kinds of edge (stick) are described. The major types of these structures are identified and reported with optimal embeddings. 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.