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AbstractWe consider smooth flows with constant slope on the 2-dimensional (flat) torus and on the (flat) Klein bottle. It is well known that for a flow with constant slope on the torus either all orbits are periodic or all orbits are dense. We obtain a similar classification for the orbits of a flow with constant slope on the Klein bottle: again either all orbits are periodic, although some periods change, or all orbits are dense. This requires considering flows on the unit normal bundle of the Klein bottle. [ABSTRACT FROM AUTHOR]

We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e. a graph Γ in the Klein bottle K. Let Γmn denote the graph obtained by pasting m rows and n columns of copies of Γ, which embeds in K for n odd and in the torus T2 for n even. We compute the dimer partition function Zmn of Γmn for n odd, in terms of the well-known characteristic polynomial P of Γ12⊂T² together with a new characteristic polynomial R of Γ⊂K. Using this result together with work of Kenyon, Sun and Wilson [arXiv:1310.2603], we show that in the bipartite case, this partition function has the asymptotic expansion logZmn=mnf0/2+fsc+o(1), for m,n tending to infinity and m/n bounded below and above, where f0 is the bulk free energy for Γ12⊂T² and fsc an explicit finite-size correction term. The remarkable feature of this later term is its universality: it does not depend on the graph Γ, but only on the zeros of P on the unit torus and on an explicit (purely imaginary) 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 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. [ABSTRACT FROM AUTHOR]

Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I, it was shown that graphs that are critical for embeddings into surfaces of Euler genus k or for embeddings into nonorientable surface of genus k are built from 3-connected components, called hoppers and cascades. In Part II, all cascades for Euler genus 2 are classified. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained. [ABSTRACT FROM AUTHOR]

Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I we consider the structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the case when Euler genus is small. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained. This is the first complete result about obstructions for embeddability of graphs in the Klein bottle, and the outcome is somewhat surprising in the sense that there are considerably fewer excluded minors than expected. [ABSTRACT FROM AUTHOR]

In this series of papers, we study the crosscap two embedding of a class of multipartite graphs, namely, annihilating-ideal graphs of a lattice. In Part 1 of the series [Class of crosscap two graphs arising from lattices-Ⅰ, Mathematics, 11 (2023), 1-26], we classified lattices with the number of atoms less than or equal to 4, whose annihilating-ideal graph can be embedded in the Klein bottle. In this paper, which is Part 2 of the series, we classify all finite lattices with at least 5 atoms whose annihilating-ideal graph is embedded in crosscap two surfaces. These characterizations help us to identify classes of multipartite graphs, which are embedded in the Klein bottle.

In this series of papers, we study the crosscap two embedding of a class of multipartite graphs, namely, annihilating-ideal graphs of a lattice. In Part 1 of the series [Class of crosscap two graphs arising from lattices-I, Mathematics, 11 (2023), 1-26], we classified lattices with the number of atoms less than or equal to 4, whose annihilating-ideal graph can be embedded in the Klein bottle. In this paper, which is Part 2 of the series, we classify all finite lattices with at least 5 atoms whose annihilating-ideal graph is embedded in crosscap two surfaces. These characterizations help us to identify classes of multipartite graphs, which are embedded in the Klein bottle. [ABSTRACT FROM AUTHOR]

The Metric of a graph plays an essential role in the arrangement of different dimensional structures and finding their basis in various terms. The metric dimension of a graph is the selection of the minimum possible number of vertices so that each vertex of the graph is distinctively defined by its vector of distances to the set of selected vertices. This set of selected vertices is known as the metric basis of a graph. In applied mathematics or computer science, the topic of metric basis is considered as locating number or locating set, and it has applications in robot navigation and finding a beacon set of a computer network. Due to the vast applications of this concept in computer science, optimization problems, and also in chemistry enormous research has been conducted. To extend this research to a four-dimensional structure, we studied the metric basis of the Klein bottle and proved that the Klein bottle has a constant metric dimension for the variation of all its parameters. Although the metric basis is variying in 3 and 4 values when the values of its parameter change, it remains constant and unchanged concerning its order or number of vertices. The methodology of determining the metric basis or locating set is based on the distances of a graph. Therefore, we proved the main theorems in distance forms. [ABSTRACT FROM AUTHOR]

Area-minimizing unit vector fields on antipodally punctured unit 2-sphere are defined in Fabiano et al. (Manuscr Math 161(3-4):487–499, 2019). For these unit vector fields with even Poincaré indexes, we prove that the topological closure of their image coincides with the image of minimally immersed Klein bottles in T 1 S 2 . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let G be a connected graph with at least 2 (m + n + 1) vertices that contains a perfect matching. Then G is E (m , n) if for each pair of disjoint matchings M , N ⊆ E (G) of size m and n , respectively, there exists a perfect matching F in G such that M ⊆ F and F ∩ N = ∅. A graph G is 1-embeddable in a surface Σ if G can be drawn in Σ so that every edge of G crosses at most one other edge at a point. R.E.L. Aldred and M.D. Plummer [1,2] investigated the properties E (m , n) for graphs embedded in the plane, torus, projective plane and Klein bottle. In this paper, we study the property E (m , n) for 1-embeddable graphs in surfaces with small genus. It is shown that no 1-embeddable graph in the plane or projective plane is E (4 , 1) and no 1-embeddable graph in the torus or Klein bottle is E (5 , 1). As corollaries, no 1-embeddable graph in the plane or projective plane is 5-extendable and no 1-embeddable graph in the torus or Klein bottle is 6-extendable. Some examples show that such results are best possible. [ABSTRACT FROM AUTHOR]

The work is devoted to the study of the deformation of one-sided sur­faces. Let a normal vector be drawn along a closed curve on the surface. If, when returning to the original point, the direction of the normal coin­cides with the original direction of the normal, then the surface is called two-sided. Otherwise, we have a one-sided surface. Unilateral surfaces include: crossed cap, Roman surface, Boya surface, Klein bottle. Roman surface, Boya surface and crossed hood are a model of the projective plane. It is proved that if the surface is a model of a Moebius strip, of a Klein bottle, of projective plane, then the surface deformation is a Moebius strip model, a Klein bottle model, projective plane model respectively. Using a mathematical package, graphs are built the surfaces under consideration.

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.

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, Möbius strips, and so forth. A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic 0, and particularly the Klein bottle associated with the square torus, whose eigenvalues have higher multiplicities. In this note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottle associated with the square torus (resp. with square fundamental domain) are the first and second eigenvalues. We also consider the flat cylinders (0,π) × S1r where r ∈ {0.5,1} is the radius of the circle S1r, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues. [ABSTRACT FROM AUTHOR]

A map M is called an almost semi-equivelar if all the vertices of M have same type face-cycle except one. The maps on the surfaces of square pyramid and the pentagonal pyramid (2 out of 92 Johnson solids) provide almost semi-equivelar maps on the sphere. In this paper, for the first time, we study and classify an almost semi-equivelar map on close surfaces, other than sphere, particularly on torus and Klein bottle on at most 15 vertices. [ABSTRACT FROM AUTHOR]

The Klein bottle and hairy ball theorem are important concepts in advanced mathematics and they are both examples of the Poincaré–Hopf theorem. Complex theories such as these usually remain unrealized in the minds of mathematicians. In this collaborative work between a mathematician and a designer, we introduce the concept of the hairy Klein bottle, which is a fusion that explores a continuous non-vanishing vector field on a non-orientable surface. We strive to explain and shape these esoteric concepts accurately and to and describe the modelling process of creating a tangible, touchable sculpture designed to intrigue and invite simple understanding. [ABSTRACT FROM AUTHOR]

We focus on the quantitative and local topological properties of range images. We consider the spaces Mm of m × m high-contrast patches of range images for m= 3, 5, 7, 9, 11. Using computational topological tools to analyze range image patches, we detect that M3 (M9, M11) has core subsets with the topology of a circle, M3, M5, M7, M9 and that M11 have some subspaces with the topology of a Klein bottle. We also discover that the largest subspace with the Klein bottle's topology decreases as the measurements of patches increase, which generalizes the results in the paper of H. Adams and G. Carlsson, and demonstrates properties among optical images and range image patches, which are more similar than those established by Lee et al. [ABSTRACT FROM AUTHOR]

The geodesic complexity of a metric space X is the smallest k for which there is a partition of X × X into locally compact sets E0, . . ., Ek on each of which there is a continuous choice of minimal geo-desic σ(x0, x1) from x0 to x1. We prove that the geodesic complexity of an n-dimensional Klein bottle Kn equals 2n. The topological complexity of Kn remains unknown for n > 2. [ABSTRACT FROM AUTHOR]

Let M be the exterior of a hyperbolic link K ∪ L in a homology 3-sphere Y, such that the linking number lk(K;L) is non-zero. In this note we prove that if γ is a slope in ∂N(L) such that the manifold ML(γ) obtained by -Dehn filling along ∂N(L) contains a Klein bottle, then there is a bound on Δ(µ; γ), depending on the genus of K and on lk(K;L). [ABSTRACT FROM AUTHOR]

Being multidisciplinary, digitally adept and small is the key to success of Jesse Louis‐Rosenberg and Jessica Rosenkrantz's generative design studio Nervous System in upstate New York that works at the intersection of science, art and technology. Their projects are inspired by natural phenomena and at once they are biological researchers, artists and coders. This synthesis of expertise and practice creates beautiful and thoughtful objects. [ABSTRACT FROM AUTHOR]

An n -dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space. [ABSTRACT FROM AUTHOR]

Let K be a hyperbolic knot in the 3 -sphere. If a p / q -Dehn surgery on K produces manifold with an embedded Klein bottle or essential 2 -torus, then we prove that | p | ≤ 8 g K − 3 , where g K is the genus of K. We obtain different upper bounds according to the production of a Klein bottle, a non-separating 2 -torus, or an essential and separating 2 -torus. The well known examples which are the figure eight knot and the pretzel knot K (− 2 , 3 , 7) reach the given upper bounds. We study this problem considering null-homologous hyperbolic knots in compact, orientable and closed 3 -manifolds. [ABSTRACT FROM AUTHOR]

In the current study, distance-based topological invariants, namely the Wiener number and the topological roundness index, were computed for graphenic tori and Klein bottles (named toroidal and Klein bottle fullerenes or polyhexes in the pre-graphene literature) described as closed graphs with N vertices and 3N/2 edges, with N depending on the variable length of the cylindrical edge LC of these nano-structures, which have a constant length LM of the Möbius zigzag edge. The presented results show that Klein bottle cubic graphs are topologically indistinguishable from toroidal lattices with the same size (N, LC, LM) over a certain threshold size LC. Both nano-structures share the same values of the topological indices that measure graph compactness and roundness, two key topological properties that largely influence lattice stability. Moreover, this newly conjectured topological similarity between the two kinds of graphs transfers the translation invariance typical of the graphenic tori to the Klein bottle polyhexes with size LC ≥ LC, making these graphs vertex transitive. This means that a traveler jumping on the nodes of these Klein bottle fullerenes is no longer able to distinguish among them by only measuring the chemical distances. This size-induced symmetry transition for Klein bottle cubic graphs represents a relevant topological effect influencing the electronic properties and the theoretical chemical stability of these two families of graphenic nano-systems. The present finding, nonetheless, provides an original argument, with potential future applications, that physical unification theory is possible, starting surprisingly from the nano-chemical topological graphenic space; thus, speculative hypotheses may be drawn, particularly relating to the computational topological unification (that is, complexification) of the quantum many-worlds picture (according to Everett’s theory) with the space-curvature sphericity/roundness of general relativity, as is also currently advocated by Wolfram’s language unification of matter-physical phenomenology.