Isotopy classes for 3-periodic net embeddings

Power, S. C.; Baburin, Igor A.; Proserpio, Davide M.

doi:10.1107/s2053273320000625

Cited by 6 publications

(11 citation statements)

References 48 publications

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“…[cf. Power et al (2020), Proposition 4.5.] Let À be a Cayley graph of Z n with respect to a generating set S, and let À be embedded in R n as described above with edges as straight-line segments.…”

Section: Theoretical Background and Computational Methodology 21 Somentioning

confidence: 99%

“…Fischer, 1974Fischer, , 1993, their potential has never been used in full [some applications are described by Eon (2012)]. Despite some results on lattice nets or bouquet nets (Delgado-Friedrichs & O'Keeffe, 2009;Moreira de Oliveira & Eon, 2014), the terms adopted by crystallographers for Cayley graphs of Z n , complete enumerations for Z 3 (under fairly natural assumptions) have become available only recently (Power et al, 2020). Many important properties of Cayley graphs of Z n were derived by Kostousov (2007) which we quote below (Section 2.1).…”

Section: Introductionmentioning

confidence: 99%

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On Cayley graphs of {\bb Z}^4

Baburin

2020

Acta Cryst Sect A

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The generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components −1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently large finite ball cut out from an infinite graph. Six exceptional, rather `dense' graphs have been identified which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10. They could be built by either interconnecting interpenetrated three- or four-dimensional cubic lattices and therefore necessarily contain Hopf links between quadrangular cycles. As a consequence, a local combinatorial isomorphism does not extend to a local isotopy.

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Section: Theoretical Background and Computational Methodology 21 Somentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Cayley graphs of {\bb Z}^4

Baburin

2020

Acta Cryst Sect A

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“…While the RCSR database contains information on 3000+ named nets, ToposPro is a more recent research tool for the geometrical and topological analysis of crystal structures with a database of more than 190 000 nets (http://topcryst.com and Blatov et al, 2014). These rapid developments in net theory provide the context for the work by Power et al (2020).…”

Section: Isotopy Classification Of Three-dimensional Embedded Netsmentioning

confidence: 99%

“…Embedded nets can have multiple connected components, which can have different dimensionality and be intertwined in rather sophisticated ways. The article by Power et al (2020) mainly concerns the behavior of entangled embedded nets with multiple connected components, as well as of self-entangled connected embedded nets, with regards to continuous deformations that avoid edge collisions or may satisfy some other desirable property (e.g. retaining some translational symmetry across the deformation path as formalized by the notion of periodic isotopy).…”

Section: Isotopy Classification Of Three-dimensional Embedded Netsmentioning

confidence: 99%

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Isotopy classification of three-dimensional embedded nets

Schulte

2020

Acta Cryst Sect A

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Nets are important mathematical tools that have been used as models in the analysis and design of crystal structures since the early days of crystallography (Wells, 1977). The notion of a net is extremely versatile and lends itself to major applications in crystal engineering and crystal structure prediction, including metal-organic frameworks (MOFs) and zeolites. Apart from this wide range of applicability, nets and their embeddings are also fascinating mathematical objects in their own right and have significant applications in mathematics itself, including in polyhedral geometry, tiling theory, minimal surfaces and discrete symmetry, to name a few.The underlying topological structure of a net is provided by an abstract graph G, with finite or (usually) countably infinite vertex set and edge set, and in the infinite case equipped with an abstract periodicity structure determined by a free abelian subgroup of the combinatorial automorphism group of G. The infinite graphs come to life through their 3-periodic embeddings as periodic bond-node structures N in an ambient geometric space, which in most applications is ordinary Euclidean 3-space R 3 . These embedded nets N can have several different characteristics. In the embedding, the vertices of G are represented by (usually distinct) points, called the vertices (or nodes) of N , and the edges of G are represented by what are called edges (or bonds) of N , which usually are straight line segments (sometimes with a specified length), possibly allowed to collide, or sometimes are non-self-intersecting paths of various sorts. The entire embedded net N is required to be invariant under a group of translations T of the ambient space generated by the translations in the directions of the basis vectors from a vector-space basis, called the periodicity basis of N . This translation group T provides a geometric representation of the abstract periodicity structure on the underlying structure graph. In a proper linear 3-periodic net N in R 3 , the edges of N are noncolliding straight line segments joining vertices of N . In applications, the vertices of an embedded net N may represent spatial positions of atoms or of clusters of atoms, such as single metal ions or secondary building units, and the edges may represent chemical bonds or ligands. A net can represent many compounds.My own interest in nets relates to discrete geometry and was inspired by the pioneering contributions of Michael O'Keeffe and his collaborators (see, for example, Delgado-Friedrichs et al., 2005). These included Omar Yaghi, who was a co-founder of reticular chemistry with O'Keeffe (for which they were jointly awarded the Gregori Aminoff Prize in Crystallography by The Royal Swedish Academy of Sciences in 2019), and Davide Proserpio, to whom I am grateful for making me aware of the extensive works on crystal nets and related geometric structures in the chemistry literature (and for kindly providing the two figures). Several databases have been created to keep track of the vast amount of information on ...

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Symmetric tangled Platonic polyhedra

Hyde

Evans

2022

Proc. Natl. Acad. Sci. U.S.A.

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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 Ih, Oh, and Td). 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 “θz” 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.

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Isotopy classes for 3-periodic net embeddings

Cited by 6 publications

References 48 publications

On Cayley graphs of {\bb Z}^4

On Cayley graphs of {\bb Z}^4

Isotopy classification of three-dimensional embedded nets

Symmetric tangled Platonic polyhedra

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