All toroidal embeddings of polyhedral graphs in 3-space are chiral

Castle, Toen; Evans, Myfanwy E.; Hyde, Stephen T.

doi:10.1039/b907338h

Cited by 16 publications

(31 citation statements)

References 16 publications

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“…Castle et al [24] proved that polyhedral toroidal molecules which contain a nontrivial knot are chiral. The chirality of polyhedral toroidal molecules which contain a nonsplit link is shown in [28].…”

Section: Theorem 1 (Existence Of Knots and Links) Let G Be An Abstracmentioning

confidence: 99%

“…Theorem 2 (Chirality [24,28]) Let G be a simple 3-connected abstractly planar graph and f : G → T 2 ⊂ R 3 be an embedding of G with image G on the torus T 2 . If G ⊂ T 2 is nonplanar embedded, then G is topologically chiral in R 3 .…”

Section: Theorem 1 (Existence Of Knots and Links) Let G Be An Abstracmentioning

confidence: 99%

“…A graph is simple if it has neither multiple edges between a given pair of vertices nor loops from a vertex to itself.) Conjecture 1 (Castle et al [24]) All polyhedral toroidal molecules contain a nontrivial knot or a nonsplit link. The main result (Theorem 1) proves this conjecture without assuming 3-connectivity or simpleness using topological graph theory.…”

Section: Introductionmentioning

confidence: 99%

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Toroidal embeddings of abstractly planar graphs are knotted or linked

Barthel

Buck

2015

J Math Chem

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Section: Theorem 1 (Existence Of Knots and Links) Let G Be An Abstracmentioning

confidence: 99%

Section: Theorem 1 (Existence Of Knots and Links) Let G Be An Abstracmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Toroidal embeddings of abstractly planar graphs are knotted or linked

Barthel

Buck

2015

J Math Chem

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“…As a result, the ideal tangled polyhedra adopt symmetries carried over from those of the underlying ideal knots. We note that the most symmetric embeddings of tangled tetrahedron and cube isotopes, which are inevitably chiral [13], are of lower symmetry than their untangled (Platonic) versions. A separate analysis reveals that toroidal tetrahedron and cube isotopes can have, at most, rotational symmetry of order 2 and 4, respectively [22].…”

Section: (C) Cube Graphsmentioning

confidence: 99%

Ideal geometry of periodic entanglements

2015

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Three-dimensional entanglements, including knots, knotted graphs, periodic arrays of woven filaments and interpenetrating nets, form an integral part of structure analysis because they influence various physical properties. Ideal embeddings of these entanglements give insight into identification and classification of the geometry and physically relevant configurations in vivo. This paper introduces an algorithm for the tightening of finite, periodic and branched entanglements to a least energy form. Our algorithm draws inspiration from the ShrinkOn-No-Overlaps (SONO) (Pieranski 1998 In Ideal knots (eds A Stasiak, V Katritch, LH Kauffman), vol. 19, pp. 20-41.) algorithm for the tightening of knots and links: we call it Periodic-Branched Shrink-On-No-Overlaps (PB-SONO). We reproduce published results for ideal configurations of knots using PB-SONO. We then examine ideal geometry for finite entangled graphs, including θ-graphs and entangled tetrahedron-and cube-graphs. Finally, we compute ideal conformations of periodic weavings and entangled nets. The resulting ideal geometry is intriguing: we see spontaneous symmetrisation in some cases, breaking of symmetry in others, as well as configurations reminiscent of biological and chemical structures in nature.

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“…5. 14) Since the area per 2223 fundamental domain is π 3 and each domain contains a half-edge, E 2d = 3(arccosh(3)) 2 π ≈…”

Section: Untangled Embeddings Of Non-planar Infinite Crystallographicmentioning

confidence: 99%

Entanglement of Embedded Graphs

Castle

Evans

Hyde

2011

Prog. Theor. Phys. Suppl.

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We discuss the identification of untangled graph embeddings for finite planar and nonplanar graphs as well as infinite crystallographic nets. Two parallel approaches are discussed: explicit 3-space embeddings and reticulations of 2-manifolds. 2D and 3D energies are proposed that allow ranking of (un)tangled embedding graphs. §1. Introduction Graphs, G are topological objects, composed of a collection of vertices, v i (i ∈ {1, N}) and edges, e j , associated with vertex pairs, e j (v k , v l ). For simplicity, assume that G is simple (i.e. for any {k, l}, the number of associated edges is at most one). We are interested in embeddings of topological graphs in euclidean 3-space, G, motivated by the plethora of three-dimensional chemical structures, which can be idealised as embedded graphs. The vertices and edges of these graphs correspond to atoms and chemical bonds in covalent crystals or organic molecules; in the case of supra-molecular materials such as metal-organic frameworks (MOFs) or DNA assemblies) these coincide with molecular groups and polymeric ligands or H-bonds respectively. We are interested in the effects of the graph embedding on the behaviour of the material.These issues are relevant to the physical properties of materials. For example, the hardness of physical glasses can be correlated with the rigidity in the resulting bonding network, and is therefore critically dependent on the topology of the glass network. 22) The viscosity of polymers in solution is intimately associated with the entanglement of the polymer chains within the solution. 10) Here we explore aspects of ambient isotopy of graph embeddings. Define all embeddings G of a graph G that share a common ambient isotopy as equivalent isotopes. Distinct isotopeswhich are not ambient isotopic -differ in the relative entanglement of graph edges. Definition of entanglement requires elucidation of an unentangled 'ground state' isotope, G 0 , of a graph. We propose a definition for G 0 for generic G, including infinite graphs, later in this paper. For now, we assume G is 3-connected and planar (and simple), so that it is a polyhedral graph. 11) In this case, G 0 necessarily embeds in the sphere (S 2 ) without edge crossings. This is also sufficient to characterise G 0 , since Whitney's theorem ensures that the 2-cell embedding of G into S 2 -the isotope G 0 -is unique. 24) Downloaded from https://academic.oup.com/ptps/article-abstract

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All toroidal embeddings of polyhedral graphs in 3-space are chiral

Cited by 16 publications

References 16 publications

Toroidal embeddings of abstractly planar graphs are knotted or linked

Toroidal embeddings of abstractly planar graphs are knotted or linked

Ideal geometry of periodic entanglements

Entanglement of Embedded Graphs

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