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Symmetry, Volume 4, Issue 1 (March 2012), Pages 1-264

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Research

Open AccessArticle Convex-Faced Combinatorially Regular Polyhedra of Small Genus
Symmetry 2012, 4(1), 1-14; doi:10.3390/sym4010001
Received: 28 November 2011 / Revised: 15 December 2011 / Accepted: 19 December 2011 / Published: 28 December 2011
Cited by 2 | PDF Full-text (271 KB)
Abstract
Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E3 of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and [...] Read more.
Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E3 of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 ≤ g ≤ 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 ≤ g ≤ 6, admitting realizations as convex-faced polyhedra in E3. For all admissible maps in this range, save Gordan’s map of genus 4, and its dual, we rule out realizability by a polyhedron in E3. Full article
(This article belongs to the Special Issue Polyhedra)
Open AccessArticle Towards Symmetry-Based Explanation of (Approximate) Shapes of Alpha-Helices and Beta-Sheets (and Beta-Barrels) in Protein Structure
Symmetry 2012, 4(1), 15-25; doi:10.3390/sym4010015
Received: 22 December 2011 / Revised: 6 January 2012 / Accepted: 12 January 2012 / Published: 19 January 2012
Cited by 1 | PDF Full-text (196 KB)
Abstract
Protein structure is invariably connected to protein function. There are two important secondary structure elements: alpha-helices and beta-sheets (which sometimes come in a shape of beta-barrels). The actual shapes of these structures can be complicated, but in the first approximation, they are [...] Read more.
Protein structure is invariably connected to protein function. There are two important secondary structure elements: alpha-helices and beta-sheets (which sometimes come in a shape of beta-barrels). The actual shapes of these structures can be complicated, but in the first approximation, they are usually approximated by, correspondingly, cylindrical spirals and planes (and cylinders, for beta-barrels). In this paper, following the ideas pioneered by a renowned mathematician M. Gromov, we use natural symmetries to show that, under reasonable assumptions, these geometric shapes are indeed the best approximating families for secondary structures. Full article
(This article belongs to the Special Issue Symmetry Group Methods for Molecular Systems)
Open AccessArticle Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori
Symmetry 2012, 4(1), 26-38; doi:10.3390/sym4010026
Received: 14 November 2011 / Revised: 12 January 2012 / Accepted: 13 January 2012 / Published: 20 January 2012
Cited by 1 | PDF Full-text (192 KB)
Abstract
A symmetry group of a spatial graph Γ in S3 is a finite group consisting of orientation-preserving self-diffeomorphisms of S3 which leave Γ setwise invariant. In this paper, we show that in many cases symmetry groups of Γ which agree on [...] Read more.
A symmetry group of a spatial graph Γ in S3 is a finite group consisting of orientation-preserving self-diffeomorphisms of S3 which leave Γ setwise invariant. In this paper, we show that in many cases symmetry groups of Γ which agree on a regular neighborhood of Γ are equivalent up to conjugate by rational twists along incompressible spheres and tori in the exterior of Γ. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Open AccessArticle Knots on a Torus: A Model of the Elementary Particles
Symmetry 2012, 4(1), 39-115; doi:10.3390/sym4010039
Received: 16 November 2011 / Revised: 27 December 2011 / Accepted: 16 January 2012 / Published: 9 February 2012
PDF Full-text (7529 KB) | HTML Full-text | XML Full-text
Abstract
Two knots; just two rudimentary knots, the unknot and the trefoil. That’s all we need to build a model of the elementary particles of physics, one with fermions and bosons, hadrons and leptons, interactions weak and strong and the attributes of spin, [...] Read more.
Two knots; just two rudimentary knots, the unknot and the trefoil. That’s all we need to build a model of the elementary particles of physics, one with fermions and bosons, hadrons and leptons, interactions weak and strong and the attributes of spin, isospin, mass, charge, CPT invariance and more. There are no quarks to provide fractional charge, no gluons to sequester them within nucleons and no “colors” to avoid violating Pauli’s principle. Nor do we require the importation of an enigmatic Higgs boson to confer mass upon the particles of our world. All the requisite attributes emerge simply (and relativistically invariant) as a result of particle conformation and occupation in and of spacetime itself, a spacetime endowed with the imprimature of general relativity. Also emerging are some novel tools for systemizing the particle taxonomy as governed by the gauge group SU(2) and the details of particle degeneracy as well as connections to Hopf algebra, Dirac theory, string theory, topological quantum field theory and dark matter. One exception: it is found necessary to invoke the munificent geometry of the icosahedron in order to provide, as per the group “flavor” SU(3), a scaffold upon which to organize the well-known three generations—no more, no less—of the particle family tree. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Open AccessArticle Defining the Symmetry of the Universal Semi-Regular Autonomous Asynchronous Systems
Symmetry 2012, 4(1), 116-128; doi:10.3390/sym4010116
Received: 1 November 2011 / Revised: 8 February 2012 / Accepted: 9 February 2012 / Published: 15 February 2012
PDF Full-text (359 KB)
Abstract
The regular autonomous asynchronous systems are the non-deterministic Boolean dynamical systems and universality means the greatest in the sense of the inclusion. The paper gives four definitions of symmetry of these systems in a slightly more general framework, called semi-regularity, and also [...] Read more.
The regular autonomous asynchronous systems are the non-deterministic Boolean dynamical systems and universality means the greatest in the sense of the inclusion. The paper gives four definitions of symmetry of these systems in a slightly more general framework, called semi-regularity, and also many examples. Full article
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
Open AccessArticle The 27 Possible Intrinsic Symmetry Groups of Two-Component Links
Symmetry 2012, 4(1), 129-142; doi:10.3390/sym4010129
Received: 13 January 2012 / Revised: 7 February 2012 / Accepted: 9 February 2012 / Published: 17 February 2012
PDF Full-text (313 KB)
Abstract
We consider the “intrinsic” symmetry group of a two-component link L, defined to be the image ∑(L) of the natural homomorphism from the standard symmetry group MCG(S3, L) to the product MCG(S3) × [...] Read more.
We consider the “intrinsic” symmetry group of a two-component link L, defined to be the image ∑(L) of the natural homomorphism from the standard symmetry group MCG(S3, L) to the product MCG(S3) × MCG(L). This group, first defined by Whitten in 1969, records directly whether L is isotopic to a link L′ obtained from L by permuting components or reversing orientations; it is a subgroup of Γ2, the group of all such operations. For two-component links, we catalog the 27 possible intrinsic symmetry groups, which represent the subgroups of Γ2 up to conjugacy. We are able to provide prime, nonsplit examples for 21 of these groups; some are classically known, some are new. We catalog the frequency at which each group appears among all 77,036 of the hyperbolic two-component links of 14 or fewer crossings in Thistlethwaite’s table. We also provide some new information about symmetry groups of the 293 non-hyperbolic two-component links of 14 or fewer crossings in the table. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
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Open AccessArticle Intrinsic Symmetry Groups of Links with 8 and Fewer Crossings
Symmetry 2012, 4(1), 143-207; doi:10.3390/sym4010143
Received: 4 January 2012 / Revised: 18 January 2012 / Accepted: 31 January 2012 / Published: 20 February 2012
Cited by 4 | PDF Full-text (1600 KB)
Abstract
We present an elementary derivation of the “intrinsic” symmetry groups for links of 8 or fewer crossings. We show that standard invariants are enough to rule out all potential symmetries outside the symmetry group of the group of the link for all [...] Read more.
We present an elementary derivation of the “intrinsic” symmetry groups for links of 8 or fewer crossings. We show that standard invariants are enough to rule out all potential symmetries outside the symmetry group of the group of the link for all but one of these links and present explicit isotopies generating the symmetry group for every link. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Open AccessArticle Self-Dual, Self-Petrie Covers of Regular Polyhedra
Symmetry 2012, 4(1), 208-218; doi:10.3390/sym4010208
Received: 17 January 2012 / Revised: 21 February 2012 / Accepted: 23 February 2012 / Published: 27 February 2012
Cited by 2 | PDF Full-text (234 KB)
Abstract
The well-known duality and Petrie duality operations on maps have natural analogs for abstract polyhedra. Regular polyhedra that are invariant under both operations have a high degree of both “external” and “internal” symmetry. The mixing operation provides a natural way to build [...] Read more.
The well-known duality and Petrie duality operations on maps have natural analogs for abstract polyhedra. Regular polyhedra that are invariant under both operations have a high degree of both “external” and “internal” symmetry. The mixing operation provides a natural way to build the minimal common cover of two polyhedra, and by mixing a regular polyhedron with its five other images under the duality operations, we are able to construct the minimal self-dual, self-Petrie cover of a regular polyhedron. Determining the full structure of these covers is challenging and generally requires that we use some of the standard algorithms in combinatorial group theory. However, we are able to develop criteria that sometimes yield the full structure without explicit calculations. Using these criteria and other interesting methods, we then calculate the size of the self-dual, self-Petrie covers of several polyhedra, including the regular convex polyhedra. Full article
(This article belongs to the Special Issue Polyhedra)
Open AccessArticle Hidden Symmetries in Simple Graphs
Symmetry 2012, 4(1), 219-224; doi:10.3390/sym4010219
Received: 15 February 2012 / Revised: 23 February 2012 / Accepted: 27 February 2012 / Published: 5 March 2012
PDF Full-text (197 KB) | HTML Full-text | XML Full-text
Abstract
It is shown that each element s in the normalizer of the automorphism group Aut(G) of a simple graph G with labeled vertex set V is an Aut(G) invariant isomorphism between G and the graph obtained [...] Read more.
It is shown that each element s in the normalizer of the automorphism group Aut(G) of a simple graph G with labeled vertex set V is an Aut(G) invariant isomorphism between G and the graph obtained from G by the s permutation of Vi.e., s is a hidden permutation symmetry of G. A simple example illustrates the theory and the applied notion of system robustness for reconfiguration under symmetry constraint (RUSC) is introduced. Full article
Open AccessArticle Classical Knot Theory
Symmetry 2012, 4(1), 225-250; doi:10.3390/sym4010225
Received: 3 February 2012 / Revised: 1 March 2012 / Accepted: 1 March 2012 / Published: 7 March 2012
Cited by 3 | PDF Full-text (9080 KB)
Abstract This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handle-decompositions of knot complements. Full article
(This article belongs to the Special Issue Symmetry and Beauty of Knots)
Open AccessArticle One-Sign Order Parameter in Iron Based Superconductor
Symmetry 2012, 4(1), 251-264; doi:10.3390/sym4010251
Received: 2 March 2012 / Revised: 14 March 2012 / Accepted: 16 March 2012 / Published: 21 March 2012
Cited by 71 | PDF Full-text (2311 KB) | HTML Full-text | XML Full-text
Abstract
The onset of superconductivity at the transition temperature is marked by the onset of order, which is characterized by an energy gap. Most models of the iron-based superconductors find a sign-changing (s±) order parameter [1–6], with the physical implication that pairing is [...] Read more.
The onset of superconductivity at the transition temperature is marked by the onset of order, which is characterized by an energy gap. Most models of the iron-based superconductors find a sign-changing (s±) order parameter [1–6], with the physical implication that pairing is driven by spin fluctuations. Recent work, however, has indicated that LiFeAs has a simple isotropic order parameter [7–9] and spin fluctuations are not necessary [7,10], contrary to the models [1–6]. The strength of the spin fluctuations has been controversial [11,12], meaning that the mechanism of superconductivity cannot as yet be determined. We report the momentum dependence of the superconducting energy gap, where we find an anisotropy that rules out coupling through spin fluctuations and the sign change. The results instead suggest that orbital fluctuations assisted by phonons [13,14] are the best explanation for superconductivity. Full article
(This article belongs to the Special Issue Symmetries of Electronic Order)
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