Klein Bottle Research Articles

Porphyro’s ‘Ancient Ditty’: Text and Topology in The Eve of St Agnes/‘La Belle Dame sans Merci’

Abstract This article explores non-orientable textualities and self-quotation in Romantic poetry, focusing on two of John Keats’s major poems: The Eve of St Agnes and ‘La Belle Dame sans Merci’. It proposes that an intricate web of intertextuality between Keats’s romance and ballad can be conceptualized in topological terms. Central to this approach is the application of the Klein bottle—a single-sided, non-orientable structure—as a model for understanding the complex ways in which the two poems are connected to themselves and each other. The analysis is extended to include the reflexive dynamics of Robert Burns’s bawdy song, ‘Nine Inch Will Please a Lady’ and Richard Roos’s Middle English translation of Alain Chartier’s ‘La Belle Dame sans Mercy’. Finally, the essay argues that Keats’s romance and ballad can be envisioned not as distinct literary works but rather as topological regions on a larger, continuous textual surface—a spatial paradigm that has the potential to provide a fresh lens for examining intertextual literature more widely.

NONEMBEDDABILITY OF THE KLEIN BOTTLE IN RP 3 AND LAWSON’S CONJECTURE

In 1985 Montiel & Ros showed that the only minimal torus in S3 , for which the first eigenvalue of the Laplacian is 2, is the Clifford torus. Here, we will show first the nonexistence of an embedded Klein bottle in RP3 . Indeed we will prove that the only non orientable closed surfaces that can be embedded in RP3 are those with odd Euler characteristic. Later on, we will give another proof of Montiel & Ros’ result, assuming that the minimal torus has {x, –x} simmetry.

Calculation of Nielsen periodic numbers of fibre preserving maps

In 2000 Heath and Keppelmann proved the naïve type addition formulae expressing the Nielsen periodic numbers NPn(f) and NFn(f) of the fibre preserving map (f,f¯) with the base map f¯n weakly Jiang, as simple sums of the Nielsen periodic numbers on the fibres. In this paper we extend these formulae to the general case. Our formulae are particularly useful for fibre preserving maps f with the base map f¯n non-weakly Jiang. We illustrate these formulae with examples of fibre preserving maps on a circle bundle over the Klein bottle.

A geometrically nonlinear Cosserat shell model for orientable and non-orientable surfaces: Discretization with geometric finite elements

We investigate discretizations of a geometrically nonlinear elastic Cosserat shell with nonplanar reference configuration originally introduced by Bîrsan, Ghiba, Martin, and Neff in 2019. The shell model includes curvature terms up to order 5 in the shell thickness, which are crucial to reliably simulate high-curvature deformations such as near-folds or creases. The original model is generalized to shells that are not homeomorphic to a subset of R2. For this, we replace the originally planar parameter domain by an abstract two-dimensional manifold, and verify that the hyperelastic shell energy and three-dimensional reconstruction are invariant under changes of the local coordinate systems. This general approach allows to determine the elastic response for even non-orientable surfaces like the Möbius strip and the Klein bottle. We discretize the model with a geometric finite element method and, using that geometric finite elements are H1-conforming, prove that the discrete shell model has a solution. Numerical tests then show the general performance and versatility of the model and discretization method.

Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary

Let $\mathcal{G}$ be a Morse-Bott foliation on the solid Klein bottle $\mathbf{K}$ into $2$-dimensional Klein bottles parallel to the boundary and one singular circle $S^1$. Let also $S^1\widetilde{\times}S^2$ be the twisted bundle over $S^1$ which is a union of two solid Klein bottles $\mathbf{K}_0$ and $\mathbf{K}_1$ with common boundary $K$. Then the above foliation $\mathcal{G}$ on both $\mathbf{K}_0$ and $\mathbf{K}_1$ gives a foliation $\mathcal{G}'$ on $S^1\widetilde{\times}S^2$ into parallel Klein bottles and two singluar circles. The paper computes the homotopy types of groups of foliated (sending leaves to leaves) and leaf preserving diffeomorphisms for foliations $\mathcal{G}$ and $\mathcal{G}'$.

Minimal nondegenerate extensions

We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension, and hence that every pseudo-unitary super modular tensor category admits a minimal modular extension. This completes the program of characterizing minimal nondegenerate extensions of braided fusion categories. Our proof relies on the new subject of fusion 2 2 -categories. We study in detail the Drinfel’d centre Z ( M o d - B ) \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) of the fusion 2 2 -category M o d - B {_{}\mathrm {Mod}\text {-}\mathcal {B}} of module categories of a braided fusion 1 1 -category B \mathcal {B} . We show that minimal nondegenerate extensions of B \mathcal {B} correspond to certain trivializations of Z ( M o d - B ) \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) . In the slightly degenerate case, such trivializations are obstructed by a class in H 5 ( K ( Z 2 , 2 ) ; k × ) H^5(K(\mathbb {Z}_2, 2); \mathbb {k}^\times ) and we use a numerical invariant—defined by evaluating a certain two-dimensional topological field theory on a Klein bottle—to prove that this obstruction always vanishes. Along the way, we develop techniques to explicitly compute in braided fusion 2 2 -categories which we expect will be of independent interest. In addition to the model of Z ( M o d - B ) \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) in terms of braided B \mathcal {B} -module categories, we develop a computationally useful model in terms of certain algebra objects in B \mathcal {B} . We construct an S S -matrix pairing for any braided fusion 2 2 -category, and show that it is nondegenerate for Z ( M o d - B ) \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) . As a corollary, we identify components of Z ( M o d - B ) \mathcal {Z}({_{}\mathrm {Mod}\text {-}\mathcal {B}}) with blocks in the annular category of B \mathcal {B} and with the homomorphisms from the Grothendieck ring of the Müger centre of B \mathcal {B} to the ground field.

Continuum of allosteric actions for non-amenable surface groups

AbstractLet $\Sigma $ be a closed surface other than the sphere, the torus, the projective plane or the Klein bottle. We construct a continuum of probability measure preserving ergodic minimal profinite actions for the fundamental group of $\Sigma $ that are topologically free but not essentially free, a property that we call allostery. Moreover, the invariant random subgroups we obtain are pairwise distincts.

Essential surfaces in Seifert fiber spaces with singular surfaces

Two-sided incompressible surfaces in Seifert fiber spaces with isolated singular fibers are well-understood. Frohman [3] and Rannard [6] have shown that one-sided incompressible surfaces in Seifert fiber spaces which have isolated singular fibers are either pseudo-horizontal or psuedo-vertical. We extend their result to characterise essential surfaces in Seifert fiber spaces which may have singular surfaces, i.e., in S1-foliated 3-manifolds which have fibered model neighbourhoods that are isomorphic to either a fibered solid torus or a fibered solid Klein bottle.

Sensitivity of the thermodynamics of two-dimensional systems towards the topological classes of their surfaces

Using Monte Carlo simulations we study the two-dimensional Ising model on closed manifolds of various topologies near the critical point of the planar bulk system. To this end we consider triangular, square, and hexagonal lattices. We find that the corresponding universal scaling functions of the magnetic susceptibility χ, as well as those of the specific heat C, differ for distinct Euler characteristics K of the manifold. In particular, for each lattice decoration the maxima of the scaling functions of χ grow as K increases. For the specific heat this relationship is inverted: e.g., for each lattice decoration the maxima of the scaling functions of C on the spherical surface (Euler characteristic K=2) is smaller than on the projective plane (K=1) which, in turn, is smaller than on the torus and on the Klein bottle (both with K=0). We find that if the aspect ratio of the lattices is conform to a square geometrical shape, the magnetic susceptibility scaling functions for different lattice types differ only by a non-universal amplitude.

Non-orientable order and non-commutative response in frustrated metamaterials.

From atomic crystals to animal flocks, the emergence of order in nature is captured by the concept of spontaneous symmetry breaking1-4. However, this cornerstone of physics is challenged when broken symmetry phases are frustrated by geometrical constraints. Such frustration dictates the behaviour of systems as diverse as spin ices5-8, confined colloidal suspensions9 and crumpled paper sheets10. These systems typically exhibit strongly degenerated and heterogeneous ground states and hence escape the Ginzburg-Landau paradigm of phase ordering. Here, combining experiments, simulations and theory we uncover an unanticipated form of topological order in globally frustrated matter: non-orientable order. We demonstrate this concept by designing globally frustrated metamaterials that spontaneously break a discrete [Formula: see text] symmetry. We observe that their equilibria are necessarily heteregeneous and extensively degenerated. We explain our observations by generalizing the theory of elasticity to non-orientable order-parameter bundles. We show that non-orientable equilibria are extensively degenerated due to the arbitrary location of topologically protected nodes and lines where the order parameter must vanish. We further show that non-orientable order applies more broadly to objects that are non-orientable themselves, such as buckled Möbius strips and Klein bottles. Finally, by applying time-dependent local perturbations to metamaterials with non-orientable order, we engineer topologically protected mechanical memories11-19, achieve non-commutative responses and show that they carry an imprint of the braiding of the loads' trajectories. Beyond mechanics, we envision non-orientability as a robust design principle for metamaterials that can effectively store information across scales, in fields as diverse as colloidal science8, photonics20, magnetism7 and atomic physics21.

Geometry and Laplacian on discrete magic carpets

We study several variants of the classical Sierpinski carpet (SC) fractal. The main examples we call infinite magic carpets (IMC), obtained by taking an infinite blowup of a discrete graph approximation to SC and identifying edges using torus, Klein bottle or projective plane type identifications. We use both theoretical and experimental methods. We prove estimates for the size of metric balls that are close to optimal. We obtain numerical approximations to the spectrum of the graph Laplacian on IMC and to solutions of the associated differential equations: Laplace equation, heat equation and wave equation. We present evidence that the random walk on IMC is transient, and that the full spectral resolution of the Laplacian on IMC involves only continuous spectrum. This paper is a contribution to a general program of eliminating unwanted boundaries in the theory of analysis on fractals.

Torsion Geometry 5-Fold Symmetry, Anholonomic Phases, Klein Bottle Logophysics, Chaos, Resonance: Applications Towards a Novel Paradigm for the Neurosciences and Consciousness

Torsion Geometry 5-Fold Symmetry, Anholonomic Phases, Klein Bottle Logophysics, Chaos, Resonance: Applications Towards a Novel Paradigm for the Neurosciences and Consciousness

Universal Scaling of Klein Bottle Entropy near Conformal Critical Points.

We show that the Klein bottle entropy [H.-H. Tu, Phys. Rev. Lett. 119, 261603 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.261603] for conformal field theories perturbed by a relevant operator is a universal function of the dimensionless coupling constant. The universal scaling of the Klein bottle entropy near criticality provides an efficient approach to extract the scaling dimension of lattice operators via data collapse. As paradigmatic examples, we validate the universal scaling of the Klein bottle entropy for Ising and Z_{3} parafermion conformal field theories with various perturbations using numerical simulation with continuous matrix product operator approach.

Topological indices and entropy of twisted cylinder‐nonorientable hexagonal Mobius strip

AbstractMobius strip is an infinite loop having one‐sided surface with no boundaries, also known as twisted cylinder. Möbius strips being widely used in different fields of engineering are of important nature in research. Being different in sizes and shapes, these can be visualized in Euclidean space but few cannot be. Topology of Mobius strips makes it a rare Euclidean representation of the infinite nature. Researchers expanded this concept and generalized it in the form of Klein bottles. In this article, we have derived various polynomials and respective topological indices for the Hexagonal Möbius graphs having each face as a hexagon. Also, inverse relationship between heat of formation and crystal size is developed for the calculated indices.

The dimer and Ising models on Klein bottles

We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e., on a graph \Gamma in the Klein bottle \mathcal{K} . Let \Gamma_{mn} denote the graph obtained by pasting m rows and n columns of copies of \Gamma , which embeds in \mathcal{K} for n odd and in the torus \mathbb{T}^2 for n even. We compute the dimer partition function Z_{mn} of \Gamma_{mn} for n odd in terms of the well-known characteristic polynomial P of \Gamma_{12}\subset\mathbb{T}^2 together with a new characteristic polynomial R of \Gamma\subset\mathcal{K} . Using this result together with the work of Kenyon, Sun and Wilson, we show that in the bipartite case, this partition function has the asymptotic expansion \log Z_{mn}=mn\frac{\mathbf{f}_0}{2}+\mathsf{fsc}+o(1) for m , n tending to infinity and m/n bounded below and above, where \mathbf{f}_0 is the bulk free energy for \Gamma_{12}\subset\mathbb{T}^2 and \mathsf{fsc} is an explicit finite-size correction term. The remarkable feature of this later term is its universality : it does not depend on the graph \Gamma , but only on the zeros of P on the unit torus and on an explicit (purely imaginary) conformal shape parameter. A similar expansion is also obtained in the non-bipartite case, assuming a conjectural condition on the zeros of P . We then show that this asymptotic expansion holds for the Ising partition function as well, with \mathsf{fsc} taking a particularly simple form: it vanishes in the subcritical regime, is equal to \log(2) in the supercritical regime, and to an explicit function of the shape parameter at criticality. These results are in full agreement with the conformal field theory predictions of Blöte, Cardy and Nightingale.We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e., on a graph \Gamma in the Klein bottle \mathcal{K} . Let \Gamma_{mn} denote the graph obtained by pasting m rows and n columns of copies of \Gamma , which embeds in \mathcal{K} for n odd and in the torus \mathbb{T}^2 for n even. We compute the dimer partition function Z_{mn} of \Gamma_{mn} for n odd in terms of the well-known characteristic polynomial P of \Gamma_{12}\subset\mathbb{T}^2 together with a new characteristic polynomial R of \Gamma\subset\mathcal{K} . Using this result together with the work of Kenyon, Sun and Wilson, we show that in the bipartite case, this partition function has the asymptotic expansion \log Z_{mn}=mn\frac{\mathbf{f}_0}{2}+\mathsf{fsc}+o(1) for m , n tending to infinity and m/n bounded below and above, where \mathbf{f}_0 is the bulk free energy for \Gamma_{12}\subset\mathbb{T}^2 and \mathsf{fsc} is an explicit finite-size correction term. The remarkable feature of this later term is its universality : it does not depend on the graph \Gamma , but only on the zeros of P on the unit torus and on an explicit (purely imaginary) conformal shape parameter. A similar expansion is also obtained in the non-bipartite case, assuming a conjectural condition on the zeros of P . We then show that this asymptotic expansion holds for the Ising partition function as well, with \mathsf{fsc} taking a particularly simple form: it vanishes in the subcritical regime, is equal to \log(2) in the supercritical regime, and to an explicit function of the shape parameter at criticality. These results are in full agreement with the conformal field theory predictions of Blöte, Cardy and Nightingale.

Link as a complete invariant of Morse-Smale 3-diffeomorphisms

In this paper we consider gradient-like Morse-Smale diffeomorphisms defined on the three-dimensional sphere S3 . For such diffeomorphisms, a complete invariant of topological conjugacy was obtained in the works of C. Bonatti, V. Grines, V. Medvedev, E. Pecu. It is an equivalence class of a set of homotopically non-trivially embedded tori and Klein bottles embedded in some closed 3-manifold whose fundamental group admits an epimorphism to the group Z . Such an invariant is called the scheme of the gradient-like diffeomorphism f:S3→S3 . We single out a class G of diffeomorphisms whose complete invariant is a topologically simpler object, namely, the link of essential knots in the manifold S2×S1 . The diffeomorphisms under consideration are determined by the fact that their non-wandering set contains a unique source, and the closures of stable saddle point manifolds bound three-dimensional balls with pairwise disjoint interiors. We prove that, in addition to the closure of these balls, a diffeomorphism of the class G contains exactly one nonwandering point, which is a fixed sink. It is established that the total invariant of topological conjugacy of class G diffeomorphisms is the space of orbits of unstable saddle separatrices in the basin of this sink. It is shown that the space of orbits is a link of non-contractible knots in the manifold S2×S1 and that the equivalence of links is tantamount to the equivalence of schemes. We also provide a realization of diffeomorphisms of the considered class along an arbitrary link consisting of essential nodes in the manifold S2×S1 .

Class of Crosscap Two Graphs Arising from Lattices–I

Let L be a lattice. The annihilating-ideal graph of L is a simple graph whose vertex set is the set of all nontrivial ideals of L and whose two distinct vertices I and J are adjacent if and only if I∧J=0. In this paper, crosscap two annihilating-ideal graphs of lattices with at most four atoms are characterized. These characterizations provide the classes of multipartite graphs, which are embedded in the Klein bottle.

Outside Front Cover: Volume 2 Issue 2

Interdisciplinary MaterialsVolume 2, Issue 2 p. i-i COVER IMAGEOpen Access Outside Front Cover: Volume 2 Issue 2 First published: 20 March 2023 https://doi.org/10.1002/idm2.12081AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Graphical Abstract Outside Front Cover Topological objects such as Möbius strip and Klein bottle display fantastic shapes in geometric space. Artificial acoustic metamaterials can make their band structures topological in momentum space, exhibiting novel and robust wave propagation or localization in lower dimensions. Recent studies of various topological phases associated with abnormal effects in acoustics further advance topological physics. Volume2, Issue2March 2023Pages i-i RelatedInformation

Minimally Immersed Klein Bottles in the Unit Tangent Bundle of the Unit 2-Sphere Arising from Area-Minimizing Unit Vector Fields on $${\mathbb {S}}^2\backslash \{N,S\}$$

Minimally Immersed Klein Bottles in the Unit Tangent Bundle of the Unit 2-Sphere Arising from Area-Minimizing Unit Vector Fields on $${\mathbb {S}}^2\backslash \{N,S\}$$

A Spanning Union of Cycles in Thin Cylinder, Torus and Klein Bottle Grid Graphs

In this paper, we propose an algorithm for obtaining the common transfer digraph Dm* for enumeration of 2-factors in graphs from the title, all of which have m·n vertices (m,n∈N,m≥2). The numerical data gathered for m≤18 reveal some matches for the numbers of 2-factors for different types of torus or Klein bottle. In the latter case, we conjecture that these numbers are invariant under twisting.