About Structure of Graph Obstructions for Klein Surface with 9 Vertices

The structure of the 9 vertex obstructive graphs for the nonorientable surface of the genus 2 is established by the method of j-transformations of the graphs. The problem of establishing the structural properties of 9 vertex obstruction graphs for the surface of the undirected genus 2 by the method of j-transformation of graphs is considered. The article has an introduction and 5 sections. The introduction contains the main definitions, which are illustrated, to some extent, in Section 1, which provides several statements about their properties. Sections 2 – 4 investigate the structural properties of 9 vertex obstruction graphs for an undirected surface by presenting as a j-image of several graphs homeomorphic to one of the Kuratovsky graphs and at least one planar or projective-planar graph. Section 5 contains a new version of the proof of the statement about the peculiarities of the minimal embeddings of finite graphs in nonorientable surfaces, namely, that, in contrast to oriented surfaces, cell boundaries do not contain repeated edges. Also in section 5 the other properties peculiar to embeddings of graphs to non-oriented surfaces and the main result are given. The main result is Theorem 1. Each obstruction graph H for a non-oriented surface N2 of genus 2 satisfies the following. 1. An arbitrary edge u,u = (a,b) is placed on the Mebius strip by some minimal embedding of the graph H in N3 and there exists a locally projective-planar subgraph K of the graph H \ u which satisfies the condition: (tK({a,b},N3)=1)˄(tK\u({a,b},N2)=2), where tK({a,b},N) is the number of reachability of the set {a,b} on the nonorientable surface N; 2. There exists the smallest inclusion of many different subgraphs Ki of a 2-connected graph H homeomorphic to the graph K+e, where K is a locally planar subgraph of the graph H (at least K+e is homemorphic to K5 or K3,3), which covers the set of edges of the graph H. Keywords: graph, Klein surface, graph structure, graph obstruction, non-oriented surface, Möbius strip.

EVOLUTION OF OBJECTS OF PARAMETRISM

Modern trends in parametric architecture take us away from the use of static surfaces, pushing the dynamics to the first place. A new approach to architecture is opened, based on the dynamic development of shaping. The desire to develop the most optimal ratio of strength and expended material leads to an assessment of the prospects for the use of minimal and one-sided surfaces. Using the example of varying the parameters of a onesided Klein surface, the evolution of a parametric object from a simple Mobius strip to a complex dynamic form is demonstrated. The formation of the analytical surface in various software packages was studied. The sphere of applicability of various stages of form evolution to architectural objects is established.

Extremal length and Dirichlet problem on Klein surfaces

The object of this paper is to extend the method of extremal length to Klein surfaces by solving conformally invariant extremal problems on the complex double. Within this method we define the extremal length, the extremal distance, the conjugate extremal distance, the modulus, the reduced extremal distance on a Klein surface and we study their dependences on arcs.

Groups of real genus p + 1, where p is prime

Let G be a finite group. The real genusρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. We classify the large groups of real genus p + 1, that is, the groups such that |G| ≥ 3(g - 1), where the genus action of G is on a bordered surface of genus g = p + 1. The group G must belong to one of four infinite families. In addition, we determine the order of the largest automorphism group of a surface of genus g for all g such that g = p + 1, where p is a prime.

PRIME ORDER AUTOMORPHISMS OF KLEIN SURFACES REPRESENTABLE BY ROTATIONS ON THE EUCLIDEAN SPACE

Let S be a bordered orientable Klein surface and p a prime. Assume that f is an order p automorphism of S. In this work we obtain the conditions on the topological type of (S, f) to be conformally equivalent to (S′, f′) where S′ is a bordered orientable Klein surface embedded in the Euclidean space and f′ is the restriction to S′ of a prime order rotation. Our results can allow the visualization of some Riemann surfaces automorphisms by representing them as restrictions of isometries of 𝕊4 or ℝ4. We illustrate this method with the order seven automorphisms of two famous Riemann surfaces: the Klein quartic and the Wiman surface.

DOUBLES OF KLEIN SURFACES

Historical note. A non-orientable surface of genus 2 (meaning 2 cross-caps) is popularly known as the Klein bottle. However, the term Klein surface comes from Felix Klein's book “On Riemann's Theory of Algebraic Functions and their Integrals” (1882) where he introduced such surfaces in the final chapter.

GROUPS OF EVEN REAL GENUS

Let G be a finite group. The real genusρ (G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we develop some constructions of groups of even real genus, first using the notion of a semidirect product. As a consequence, we are able to show that for each integer g in certain congruence classes, there is at least one group of genus g. Next we consider the direct product Zn × G, in which one factor is cyclic and the other is a group of odd order that is generated by two elements. By placing a restriction on the genus action of G, we find the real genus of the direct product, in case n is relatively prime to |G|. We give some applications of this result, in particular to O*-groups, the odd order groups of maximum possible order. Finally we apply our results to the problem of determining whether there is a group of real genus g for each value of g. We prove that the set of integers for which there is a group has lower density greater than 5/6.