Crystallographic descriptions of regular 2-periodic weavings of threads, loops and nets

Periodic Borromean rings, rods and chains

This article describes periodic polycatenane structures built from interlocked rings in which no two are directly linked. The 2-periodic vertex-, edge- and ring-transitive families of hexagonal Borromean rings are described in detail, and it is shown how these give rise to 1- and 3-periodic ring-transitive (isonemal) families. A second isonemal 2-periodic family is identified, as is a unique 3-periodic Borromean assembly of equilateral triangles. Also reported is a notable 2-periodic structure comprising chains of linked rings in which the chains are locked in place but no two chains are directly interlinked, being held in place as a novel `quasi-Borromean' set of four repeating components.

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.

Symmetry groups of two-way twofold and three-way threefold fabrics

This work discusses the symmetry groups of two classes of woven fabrics, two-way twofold fabrics and three-way threefold fabrics. A method to arrive at a design of a fabric is presented, employing methods in color symmetry theory. Geometric representations of all possible layer group or diperiodic symmetry structures of the fabrics are derived. There are 50 layer symmetry groups corresponding to two-way twofold fabrics and 27 layer symmetry groups corresponding to three-way threefold fabrics.

Isogonal embeddings of interwoven and self-entangled honeycomb (hcb) nets and related interpenetrating primitive cubic (pcu) nets

Two- and three-periodic vertex-transitive (isogonal) piecewise-linear embeddings of self-entangled and interwoven honeycomb nets are described. The infinite families with trigonal symmetry and edge transitivity (isotoxal) are particularly interesting as they have the Borromean property that no two nets are directly linked. These also lead directly to infinite families of interpenetrating primitive cubic nets (pcu) that are also vertex- and edge-transitive and have embeddings with 90° angles between edges.

Woven, Polycatenated, or Cage Structures: Effect of Modulation of Ligand Curvature in Heteroleptic Uranyl Ion Complexes

Combining the flexible zwitterionic dicarboxylate 4,4'-bis(2-carboxylatoethyl)-4,4'-bipyridinium (L) and the anionic dicarboxylate ligands isophthalate (ipht^2-) and 1,2-, 1,3-, or 1,4-phenylenediacetate (1,2-, 1,3-, and 1,4-pda^2-), of varying shape and curvature, has allowed isolation of five uranyl ion complexes by synthesis under solvo-hydrothermal conditions. [(UO₂)₂(L)(ipht)₂] (1) and [(UO₂)₂(L)(1,2-pda)₂]·2H₂O (2) have the same stoichiometry, and both crystallize as monoperiodic coordination polymers containing two uranyl-(anionic carboxylate) strands united by L linkers into a wide ribbon, all ligands being in the divergent conformation. Complex 3, [(UO₂)₂(L)(1,3-pda)₂]·0.5CH₃CN, with the same stoichiometry but ligands in a convergent conformation, is a discrete, binuclear species which is the first example of a heteroleptic uranyl carboxylate coordination cage. With all ligands in a divergent conformation, [(UO₂)₂(L)(1,4-pda)(1,4-pdaH)₂] (4) crystallizes as a sinuous and thread-like monoperiodic polymer; two families of chains run along different directions and are woven into diperiodic layers. Modification of the synthetic conditions leads to [(UO₂)₄(LH)₂(1,4-pda)₅]·H₂O·2CH₃CN (5), a monoperiodic polymer based on tetranuclear (UO₂)₄(1,4-pda)₄ rings; intrachain hydrogen bonding of the terminal LH⁺ ligands results in diperiodic network formation through parallel polycatenation involving the tetranuclear rings and the LH⁺ rods. Complexes 1-3 and 5 are emissive, with complex 2 having the highest photoluminescence quantum yield (19%), and their spectra show the maxima positions usual for tris-κ²O,O'-chelated uranyl cations.

Borromean rings redux. A missing link found - a Borromean triplet of Borromean triplets

This paper describes a nine-component Borromean structure - a Borromean triplet of Borromean triplets - that was missing from an earlier enumeration.

Percolation and dissolution of Borromean networks

Inspired by experiments on topologically linked DNA networks, we consider the connectivity of Borromean networks, in which no two rings share a pairwise-link, but groups of three rings form inseparable triplets. Specifically, we focus on square lattices at which each node is embedded a loop which forms a Borromean link with pairs of its nearest neighbors. By mapping the Borromean link network onto a lattice representation, we investigate the percolation threshold of these networks (the fraction of occupied nodes required for a giant component), as well as the dissolution properties: the spectrum of topological links that would be released if the network were dissolved to varying degrees. We find that the percolation threshold of the Borromean square lattice occurs when approximately 60.75% of nodes are occupied, slightly higher than the 59.27% typical of a square lattice. Compared to the dissolution of Hopf-linked networks, a dissolved Borromean network will yield more isolated loops, and fewer isolated triplets per single loop. Our simulation results may be used to predict experiments from Borromean structures produced by synthetic chemistry.

Crystallographic-Morphological Connections in Star Shaped Metal-Organic Frameworks

The symmetry of a crystal's morphology usually reflects the symmetry of the crystallographic packing. For single crystals, the space and point groups allow only a limited number of mathematical descriptions of the morphology (forms), all of which are convex polyhedrons. In contrast, concave polyhedrons are a hallmark of twinning and polycrystallinity and are typically inconsistent with single crystallinity. Here we report a new type of structure: a concave polyhedron shape single crystal having a multidomain appearance and a rare space group (P622). Despite these unusual structural features, the hexagonal symmetry is revealed at the morphological levels.

Symmetric Tangling of Honeycomb Networks

Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.

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.

Isogonal weavings on the sphere: knots, links, polycatenanes

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.