Academic literature on the topic 'Surfaces non orientables'

Silva et al. produced quantum codes related to topology and coloring, which are associated with tessellations on the orientable surfaces of genus $\ge 1$ and the non-orientable surfaces of the genus 1. Current work presents an approach to build quantum surface and color codes} on non-orientable surfaces of genus $ \geq 2n+1 $ for $n\geq 1$. We also present several tables of new surface and color codes related to non-orientable surfaces. These codes have the ratios $k/n$ and $d/n$ better than the codes obtained from orientable surfaces.

AbstractA compact hyperbolic surface of genus g is called an extremal surface if it admits an extremal disc, a disc of the largest radius determined by g. Our problem is to find how many extremal discs are embedded in non-orientable extremal surfaces. It is known that non-orientable extremal surfaces of genus g > 6 contain exactly one extremal disc and that of genus 3 or 4 contain at most two. In the present paper we shall give all the non-orientable extremal surfaces of genus 5, and find the locations of all extremal discs in those surfaces. As a consequence, non-orientable extremal surfaces of genus 5 contain at most two extremal discs.

Let Ng,n be a non-orientable surface of genus g with n punctures and one boundary component. In this paper, we describe multicurves in Ng,n making use of generalized Dynnikov coordinates, and give explicit formulae for the action of crosscap transpositions and their inverses on the set of multicurves in Ng,n in terms of generalized Dynnikov coordinates. This provides a way to solve on non-orientable surfaces various dynamical and combinatorial problems that arise in the study of mapping class groups and Thurston’s theory of surface homeomorphisms, which were solved only on orientable surfaces before.

In formulating a non-orientable analogue of the Milnor Conjecture on the 4 4 -genus of torus knots, Batson [Math. Res. Lett. 21 (2014), pp. 423–436] developed an elegant construction that produces a smooth non-orientable spanning surface in B 4 B^4 for a given torus knot in S 3 S^3 . While Lobb [Math. Res. Lett. 26 (2019), pp. 1789] showed that Batson’s surfaces do not always minimize the non-orientable 4 4 -genus, we prove that they do minimize among surfaces that share their normal Euler number. We also determine the possible pairs of normal Euler number and first Betti number for non-orientable surfaces whose boundary lies in a class of torus knots for which Batson’s surfaces are non-orientable 4 4 -genus minimizers.

The deformation theory of nonorientable surfaces deals with the problem of studying parameter spaces for the different dianalytic structures that a surface can have. It is an extension of the classical theory of Teichmüller spaces of Riemann surfaces, and as such, it is quite rich. In this paper we study some basic properties of the Teichmüller spaces of non-orientable surfaces, whose parallels in the orientable situation are well known. More precisely, we prove an uniformization theorem, similar to the case of Riemann surfaces, which shows that a non-orientable compact surface can be represented as the quotient of a simply connected domain of the Riemann sphere, by a discrete group of Möbius and anti-Möbius transformation (mappings whose conjugates are Mobius transformations). This uniformization result allows us to give explicit examples of Teichmüller spaces of non-orientable surfaces, as subsets of deformation spaces of orientable surfaces. We also prove two isomorphism theorems: in the first place, we show that the Teichmüller spaces of surfaces of different topological type are not, in general, equivalent. We then show that, if the topological type is preserved, but the signature changes, then the deformations spaces are isomorphic. These are generalizations of the Patterson and Bers-Greenberg theorems for Teichmüller spaces of Riemann surfaces, respectively.

Nous considérons l'espace de représentations Hom(Pi,G) d'un groupe de surface Pi dans un groupe de Lie G, et l'espace de modules X(Pi,G) des classes de conjugaison de ces représentations. Le groupe modulaire de la surface sous-jacente agit naturellement sur ces espaces, et cette action possède une dynamique très riche qui dépend du choix du groupe de Lie G, et de la composante connexe de l'espace sur laquelle on se place. Dans cette thèse, nous étudions le cas où S est une surface non-orientable. Dans la première partie, nous étudions les propriétés dynamiques de l'action du groupe modulaire sur l'espace de modules X(Pi, SU(2)) et prouvons que cette action est ergodique lorsque la caractéristique d'Euler de la surface est inférieure à -2. Dans la deuxième partie, nous montrons que l'espace des représentations Hom(Pi, PSL(2,R)) possède deux composantes connexes indexées par une classe de Stiefel-Whitney
We consider the space of representations Hom(Pi,G) of a surface group Pi into a Lie group G, and the moduli space X(Pi,G) of G-conjugacy classes of such representations. These spaces admit a natural action of the mapping class group of the underlying surface S, and this actions displays very rich dynamics depending on the choice of the Lie group G, and on the connected component of the space that we consider. In this thesis, we focus on the case when S is a non-orientable surface. In the rst part, we study the dynamical properties of the mapping class group actions on the moduli space X(Pi,SU(2)) and prove that this action is ergodic when the Euler characteristic of the surface is less than -1 with respect to a natural measure on the space. In the second part, we show that the representation space Hom (Pi , PSL(2,R)) has two connected components indexed by a Stiefel-Whitney class

Nous considérons l'espace de représentations Hom(Pi,G) d'un groupe de surface Pi dans un groupe de Lie G, et l'espace de modules X(Pi,G) des classes de conjugaison de ces représentations. Le groupe modulaire de la surface sous-jacente agit naturellement sur ces espaces, et cette action possède une dynamique très riche qui dépend du choix du groupe de Lie G, et de la composante connexe de l'espace sur laquelle on se place. Dans cette thèse, nous étudions le cas où S est une surface non-orientable. Dans la première partie, nous étudions les propriétés dynamiques de l'action du groupe modulaire sur l'espace de modules X(Pi, SU(2)) et prouvons que cette action est ergodique lorsque la caractéristique d'Euler de la surface est inférieure à -2. Dans la deuxième partie, nous montrons que l'espace des représentations Hom(Pi, PSL(2,R)) possède deux composantes connexes indexées par une classe de Stiefel-Whitney.

La première partie du seizième problème de Hilbert traite de la topologie des courbes algébriques réelles régulières dans le plan projectif. Il est bien connu que bon nombre des propriétés topologiques satisfaites par de telles courbes sont également vraies pour la classe plus large des courbes flexibles, introduites par O. Viro en 1984. Le but de cette thèse est d'approfondir les origines topologiques des restrictions sur les courbes réelles, en lien avec le seizième problème de Hilbert. Nous ajoutons une condition naturelle à la définition de courbe flexible, à savoir qu'elles doivent intersecter une conique réelle vide Q comme une courbe algébrique, c'est-à-dire en des points positifs uniquement. Nous voyons CP(2) comme un cylindre sur un espace lenticulaire L(4,1)×R, que l'on compactifie en ajoutant RP(2) et Q aux bords, et nous utilisons la décomposition induite sur S(4)=CP(2)/conj. C'est un fait standard que la surface d'Arnold joue un rôle essentiel dans l'étude des courbes de degré pair. Nous introduisons un analogue de cette surface pour des courbes de degré impair. Nous généralisons également la notion de courbe flexible pour inclure des surfaces non orientables. Nous considérons qu'une courbe flexible est de degré m si son auto-intersection est m² et si elle intersecte la conique Q de manière transverse en exactement 2m points. Notre résultat principal affirme que pour une telle courbe flexible (non nécessairement orientable) de degré impair m=2k+1 ne peut pas posséder plus de -χ(F)/2-k²+k+1, où χ(F) est la caractéristique d'Euler de F. Cette borne supérieure se simplifie en k² dans le cas d'une courbe flexible au sens usuel. Nous généralisons également notre résultat pour des courbes flexibles sur des quadriques, ce qui produit une nouvelle restriction, même pour des courbes algébriques. Dans les chapitres introductifs, un aperçu détaillé de la théorie classique des courbes réelles planes est fait, en s'appuyant aussi bien sur le point de vue réel que complexe. Certains résultats à propos de la théorie des surfaces nouées dans les 4-variétés sont énoncés. Plus précisément, il est question de faits concernant la classe d'Euler du fibré normal d'une surface plongée. Cela nous amène ensuite à considérer la fonction de genre non-orientable d'une 4-variété. Cela constitue un analogue de la conjecture de Thom (résolue par Kronheimer et Mrowka en 1994) pour des surfaces non orientables. Nous calculons presque totalement cette fonction pour CP(2), et nous étudions cette fonction sur d'autres 4-variétés. Enfin, nous digressons autour de la nouvelle notion de courbes flexibles non orientables, où nous dressons une liste de résultats connus qui restent vrai dans ce cadre. Nous nous concentrons aussi sur la classe des courbes algébriques et flexibles qui sont invariantes sous l'action d'une involution holomorphe de CP(2), une notion introduite par T. Fiedler et appelées courbes symétriques. Nous donnons un état de l'art, et nous formulons une succession de petits résultats à propos de la disposition d'une courbe symétrique par rapport aux éléments de symétrie. Nous proposons également une approche pour tenter de généraliser la congruence de Fiedler p-n≡k² [16], valable pour des M-courbes symétriques de degré 2k, à des (M-1)-courbes symétriques de degré 2k
The first part of Hilbert's sixteenth problem deals with the topology of non-singular real plane algebraic curves in the projective plane. As well-known, many topological properties of such curves are shared with the wider class of flexible curves, introduced by O. Viro in 1984. The goal of this thesis is to further investigate the topological origins of the restrictions on real curves in connection with Hilbert's sixteenth problem. We add a natural condition to the definition of flexible curves, namely that they shall intersect an empty real conic Q like algebraic curves do, i.e. all intersections are positive. We see CP(2) as a cylinder over a lens space L(4,1)×R which is compactified by adding RP(2) and Q respectively to the ends, and we use the induced decomposition of S(4)=CP(2)/conj. It is a standard fact that Arnold's surface plays an essential role in the study of curves of even degree. We introduce an analogue of this surface for curves of odd degree. We generalize the notion of flexible curves further to include non-orientable surfaces as well. We say that a flexible curve is of degree m if its self-intersection is m² and it intersects the conic Q transversely in exactly 2m points. Our main result states that for a not necessarily orientable curve of odd degree 2k+1, its number of non-empty ovals is no larger than χ(F)/2-k²+k+1, where χ(F) is the Euler characteristic of F. This upper bound simplifies to k² in the case of a usual flexible curve. We also generalize our result for flexible curves on quadrics, which provides a new restriction, even for algebraic curves. In the introductory chapters, a thorough survey of the classical theory of real plane curves is outlined, both from the real and the complex points of view. Some results regarding the theory of knotted surfaces in 4-manifolds are laid down. More specifically, we review statements involving the Euler class of normal bundles of embedded surfaces. This eventually leads us to consider the non-orientable genus function of a 4-manifold. This forms a non-orientable counterpart of the Thom conjecture, proved by Kronheimer and Mrowka in 1994 in the orientable case. We almost entirely compute this function in the case of CP(2), and we investigate that function on other 4-manifolds. Finally, we digress around the new notion of non-orientable flexible curves, where we survey which known results still hold in that setting. We also focus on algebraic and flexible curves invariant under a holomorphic involution of CP(2), a smaller class of curves introduced by T. Fiedler and called symmetric curves. We give a state of the art, and we formulate a collection of small results results regarding the position of a symmetric plane curve with respect to the elements of symmetry. We also propose a possible approach to generalize Fiedler's congruence p-n≡k² [16], holding for symmetric M-curves of even degree 2k, into one for symmetric (M-1)-curves of even degree

In 2002, Fomin and Zelevinsky introduced a cluster algebra; a dynamical system that has already proved to be ubiquitous within mathematics. In particular, it was shown by Fomin, Shapiro and Thurston that some cluster algebras arise from orientable surfaces. Subsequently, Dupont and Palesi extended this construction to non-orientable surfaces, giving birth to quasi-cluster algebras. The finite type cluster algebras possess the remarkable property of their exchanges graphs being polytopal. We discover that the finite type quasi-cluster algebras enjoy a closely related property, namely, their exchange graphs are spherical. Revealing yet more connections we unify these two frameworks via Lam and Pylyavskyy's Laurent phenomenon algebras, showing that both orientable and non-orientable marked surfaces have an associated LP-algebra. The integration of these structures is attempted in two ways. Firstly we show that the quasi-cluster algebras of unpunctured surfaces have LP structures. Secondly, to obtain a connection for all marked surfaces, we consider laminations, forging the notion of the laminated quasi-cluster algebra. We show that each marked surface exhibits a lamination which supplies the laminated quasi-cluster algebra with an LP structure.

We discuss a method of classifying 2-dimensional invertible topological quantum field theories (TQFTs) whose domain surface categories allow non-orientable cobordisms. These are known as Klein TQFTs. To this end we study the 1+1 dimensional open-closed unoriented cobordism category K, whose objects are compact 1-manifolds and whose morphisms are compact (not necessarily orientable) cobordisms up to homeomorphism. We are able to compute the fundamental group of its classifying space BK and, by way of this result, derive an infinite loop splitting of BK, a classification of functors K → Z, and a classification of 2-dimensional open-closed invertible Klein TQFTs. Analogous results are obtained for the two subcategories of K whose objects are closed or have boundary respectively, including classifications of both closed and open invertible Klein TQFTs. The results obtained throughout the paper are generalisations of previous results by Tillmann [Til96] and Douglas [Dou00] regarding the 1+1 dimensional closed and open-closed oriented cobordism categories. Finally we consider how our results should be interpreted in terms of the known classification of 2-dimensional TQFTs in terms of Frobenius algebras.

Abstract The primary focus of this chapter involves putting graphs on surfaces. Questions that we discuss include the following. Which graphs are planar; that is, which graphs can be drawn in the plane without any edges crossing? If a graph is not planar, what is the smallest number of crossings in any drawing of it? How many planar graphs are needed to form a given graph? In what surfaces can a non-planar graph be embedded? There are many links between graph theory and topology, but the strongest is that of drawings and embeddings of graphs on surfaces. Our survey of this area of mathematics is divided into two parts, the first (Sections 11.2-11.4) on graphs in the plane, including the topics of crossing number and thickness, and the second (Sections 11.5-11.8) on embeddings in other surfaces, both orientable and non-orientable.

‘Making surfaces’ considers the shape of surfaces and discusses the work of some of the early topologists, Möbius, Klein, and Riemann. It introduces the torus shape and shows how its Euler number can be calculated along with that of a sphere. It discusses closed surfaces—ones without a boundary—and how they can be divided up into vertices, edges, and faces. It then introduces one-sided surfaces such as the Möbius strip and Klein bottle, which are examples of non-orientable surfaces. The Euler number goes a long way to separating out different surfaces, with the only missing ingredient in the classification the notion of orientability.