Isogonal weavings on the sphere: knots, links, polycatenanes

Isogonal 2-periodic polycatenanes: chain mail

For 2-periodic polycatenanes with isogonal (vertex-transitive) embeddings, the basic units linked are torus knots and links including the unknots (untangled polygons). Twenty-four infinite families have been identified, with hexagonal, tetragonal or rectangular symmetry. The simplest members of each family are described and illustrated. A method for determining the catenation number of a ring based on electromagnetic theory is described.

Piecewise-linear embeddings of decussate extended θ graphs and tetrahedra

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.

The Symmetry and Topology of Finite and Periodic Graphs and Their Embeddings in Three-Dimensional Euclidean Space

We make the case for the universal use of the Hermann-Mauguin (international) notation for the description of rigid-body symmetries in Euclidean space. We emphasize the importance of distinguishing between graphs and their embeddings and provide examples of 0-, 1-, 2-, and 3-periodic structures. Embeddings of graphs are given as piecewise linear with finite, non-intersecting edges. We call attention to problems of conflicting terminology when disciplines such as materials chemistry and mathematics collide.

Amplification of weak chiral inductions for excellent control over the helical orientation of discrete topologically chiral (M3L2) n polyhedra

Superb control over the helical chirality of discrete (M₃L₂)_n polyhedra (n = 2,4,8, M = Cu^I or Ag^I) created from the self-assembly of propeller-shaped ligands (L) equipped with chiral side chains is demonstrated here. Almost perfect chiral induction (>99 : 1) of the helical orientation of the framework was achieved for the largest (M₃L₂)₈ cube with 48 small chiral side chains (diameter: ∼5 nm), while no or moderate chiral induction was observed for smaller polyhedra (n = 2, 4). Thus, amplification of the weak chiral inductions of each ligand unit is an efficient way to control the chirality of large discrete nanostructures with high structural complexity.

Superb control over the helical chirality of highly-entangled (M₃L₂)_n polyhedra (M = Cu(i), Ag(i); n = 2,4,8) was achieved via multiplication of weak chiral inductions by side chains accumulated on the huge polyhedral surfaces.

Isogonal piecewise-linear embeddings of 1-periodic knots and links, and related 2-periodic chain-link and knitting patterns

Families of 1- and 2-periodic knots and weavings that have isogonal (vertex-transitive) piecewise-linear embeddings are described. In these structures there is just one thread, or multiple threads with parallel or collinear axes. The principal structures are a large family of 1-periodic knots and related multi-thread infinite links, knitting patterns and chain-link weaving. The relevance to synthetic chemistry is described in terms of targets for designed synthesis such as mechanically interlocked polymers.

Tangled piecewise-linear embeddings of trivalent graphs

A method is described for generating and exploring tangled piecewise-linear embeddings of trivalent graphs under the constraints of point-group symmetry. It is shown that the possible vertex-transitive tangles are either graphs of vertex-transitive polyhedra or bipartite vertex-transitive nonplanar graphs. One tangle is found for 6 vertices, three for 8 vertices (tangled cubes), seven for 10 vertices, and 21 for 12 vertices. Also described are four isogonal embeddings of pairs of cubes and 12 triplets of tangled cubes (16 and 24 vertices, respectively). Vertex 2-transitive embeddings are obtained for tangled trivalent graphs with 6 vertices (two found) and 8 vertices (45 found). Symmetrical tangles of the 10-vertex Petersen graph and the 20-vertex Desargues graph are also described. Extensions to periodic tangles are indicated. These are all interesting and viable targets for molecular synthesis.

Symmetric tangled Platonic polyhedra

Tangled tetrahedra, octahedra, cubes, icosahedra, and dodecahedra are generalizations of classical—untangled—Platonic polyhedra. Like the Platonic polyhedra, all vertices, edges, and faces are symmetrically equivalent. However, the edges of tangled polyhedra are curvilinear, or kinked, to allow entanglement, much like warps and wefts in woven fabrics. We construct the most symmetric entanglements of these polyhedra via assemblies of multistrand helices wound around edges of the conventional polyhedra; they are all necessarily chiral. The construction gives self-entangled chiral polyhedra and compound polyhedra containing catenated multiple tetrahedra or “generalized θ-polyhedra.” An unlimited variety of tangling is possible for any given topology. Related structures have been observed in synthetic materials and clathrin assemblies within cells.

Conventional embeddings of the edge-graphs of Platonic polyhedra, {f, z}, where f, z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway’s two-dimensional (2D) orbifold notation (equivalent to Schönflies symbols I_h, O_h, and T_d). Tangled Platonic {f, z} polyhedra—which cannot lie on the sphere without edge-crossings—are constructed as windings of helices with three, five, seven,… strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the “θ_z” polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.

Piecewise-linear embeddings of knots and links with rotoinversion symmetry

This article describes the simplest members of an infinite family of knots and links that have achiral piecewise-linear embeddings in which linear segments (sticks) meet at corners. The structures described are all corner- and stick-2-transitive - the smallest possible for achiral knots.

On Borromean links and related structures

The creation of knotted, woven and linked molecular structures is an exciting and growing field in synthetic chemistry. Presented here is a description of an extended family of structures related to the classical `Borromean rings', in which no two rings are directly linked. These structures may serve as templates for the designed synthesis of Borromean polycatenanes. Links of n components in which no two are directly linked are termed `n-Borromean' [Liang & Mislow (1994). J. Math. Chem. 16, 27-35]. In the classic Borromean rings the components are three rings (closed loops). More generally, they may be a finite number of periodic objects such as graphs (nets), or sets of strings related by translations as in periodic chain mail. It has been shown [Chamberland & Herman (2015). Math. Intelligencer, 37, 20-25] that the linking patterns can be described by complete directed graphs (known as tournaments) and those up to 13 vertices that are vertex-transitive are enumerated. In turn, these lead to ring-transitive (isonemal) n-Borromean rings. Optimal piecewise-linear embeddings of such structures are given in their highest-symmetry point groups. In particular, isonemal embeddings with rotoinversion symmetry are described for three, five, six, seven, nine, ten, 11, 13 and 14 rings. Piecewise-linear embeddings are also given of isonemal 1- and 2-periodic polycatenanes (chains and chain mail) in their highest-symmetry setting. Also the linking of n-Borromean sets of interleaved honeycomb nets is described.

Isogonal piecewise linear embeddings of 1-periodic weaves and some related structures

Crystallographic descriptions of isogonal piecewise linear embeddings of 1-periodic weaves and links (chains) are presented. These are composed of straight segments (sticks) that meet at corners (2-valent vertices). Descriptions are also given of some plaits - woven periodic bands, three simple periodic knots and isogonal interwoven rods.