Isogonal weavings on the sphere: knots, links, polycatenanes

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.

On Borromean links and related structures

“…A general survey of the field of piecewiselinear interlocked structures (Lu et al, 2018) focused particularly on the 3-periodic case. We have subsequently examined in more detail minimal-transitivity 2-periodic (O'Keeffe & Treacy, 2020a) and 1-periodic links and weaves (O'Keeffe & Treacy, 2021b) and finite knots and links (O'Keeffe & Treacy, 2020b). The current paper is an extension of that work.…”

“…In our earlier work (O'Keeffe & Treacy, 2020b) we identified all finite knots and links that were corner (vertex-)-transitive in a piecewise-linear embedding. They included 36 cubic and icosahedral isotoxal links, and the infinite family of torus knots and links that are stick-2-transitive.…”

“…For weavings, the term 'isonemal' has been suggested for structures with one kind of thread (Grü nbaum & Shephard, 1980). We (O'Keeffe & Treacy, 2020b) consider links also as weavings, so we use that term to describe ring-transitive links. The term 'isotoxal' is also generally accepted for structures with one kind of edge.…”

Piecewise-linear embeddings of knots and links with rotoinversion symmetry

“…At their maximum-girth embeddings, all but B(2,3) are monotoxal. This is in contrast to the behavior of torus knots, for which the maximum girth decreases uniformly with crossing number, and monotoxal embeddings are impossible when straight-stick intersections and stick bending are prohibited (O'Keeffe & Treacy, 2020b).…”

Symmetric tangled Platonic polyhedra

“…The occurrence of tangled two- and three-periodic nets at the molecular scale in synthetic chemical materials, particularly MOFs, has led to a number of fundamental studies of tangles of infinite periodic nets ( 10 , 16 – 19 ). In contrast, surprisingly little is known about allowed symmetries of finite nets, with the exception of the “trivial” (untangled) symmetric (e.g., Platonic and Archimedean) polyhedra and studies of symmetric embeddings of knots and links ( 20 – 22 ). Graph topologies of the five Platonic polyhedral nets are given by their Schläfli symbols, { f , z }, where f describes the number of edges per face and z the number of faces per vertex ( z ), including {3, 3} (tetrahedron), {3, 4} (octahedron), {4, 3} (cube), {3, 5} (icosahedron), and {5, 3} (dodecahedron).…”