A unified framework for computing geodesic distances on nonorientable manifold polyhedral surfaces

Classification of nonorientable regular embeddings of complete bipartite graphs

A combinatorial map is a connected topological graph cellularly embedded in a surface. This monograph concentrates on the automorphism group of a map, which is related to the automorphism group of a Klein surface and a Smarandache manifold, also applied to the enumeration of unrooted maps on orientable and non-orientable surfaces. A number of results for the enumeration of unrooted maps underlying a graph on orientable and non-orientable surfaces are discovered. An elementary classification for the closed s-manifolds is found. Open problems related the combinatorial maps with the differential geometry, Riemann geometry and Smarandache geometries are also presented in this monograph for the further application of the combinatorial maps to the classical mathematics.

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper [Embeddings of *-graphs into 2-surfaces, preprint (2012), arXiv:1212.5646] by the authors, the problem of calculating whether a given *-graph in which all vertices have degree 4 or 6 admits a ℤ2-homologically trivial embedding into a given orientable surface was shown to be equivalent to a problem on matrices. Here we extend those results to nonorientable surfaces. The embeddability condition that we obtain yields quadratic-time algorithms to determine whether a *-graph with all vertices of degree 4 or 6 admits a ℤ2-homologically trivial embedding into the projective plane or into the Klein bottle.

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 this paper we will introduce the concept of canonical reducing set of a surface homeomorphism, and prove that it is unique up to an isotopy. As an application, we will give a simple proof of Thurston's theorem on classifying mappings on non-orientable surface, using the techniques of quasiconformal mappings and some known results in the orientable case, especially the Thurston theorem on orientable surfaces.

The total embedding polynomial of a graph G is the bivariate polynomial where is the number of embeddings, for into the orientable surface , and is the number of embeddings, for into the nonorientable surface . The sequence is called the total embedding distribution of the graph G; it is known for relatively few classes of graphs, compared to the genus distribution . The circular ladder graph is the Cartesian product of the complete graph on two vertices and the cycle graph on n vertices. In this article, we derive a closed formula for the total embedding distribution of circular ladders.

The genus of a random graph

This chapter presents a general-purpose definition of discrete curves, surfaces, and manifolds. This definition only refers to a simple graph, \(G=(V,E)\) and its topological structure. Similar to digital manifolds defined in Chap. 5, the ideas presented in this chapter still use recursive definitions for discrete curves, surfaces, solid objects, and so on. Specifically, a vertex is a point-cell, and an edge is a line-cell. A surface-cell (2-cell) is defined as a special simple closed “curve”—a closed semi-curve with the minimum cycle property. In general, a k-cell will be a closed semi \((k-1)\)-manifold with minimum “cycle” property. For a graph G, all i-cells, \(i=0,...,n+1\), will provide topological structure to the discrete space G. An n dimensional discrete manifold M is defined as: (1) M consists of n-cells and any two n-cells are \((n-1)\)-dimensionally connected, (2) each \((n-1)\)-cell in M is contained by one or two n-cells, and (3) there is no \((n+1)\)-cell in M. We also consider the definition of orientable and non-orientable surfaces with a corresponding decision procedure. Finally, some unconventional examples of the definitions, such as quadtree surface-cell representation, an octree solid-cell representation, Voronoi decomposition, and Delaunay simplifications are presented.

Shape deformations preserving the intrinsic properties of a surface are called isometries. An isometry deforms a surface without tearing or stretching it, and preserves geodesic distances. We present a technique for matching point set surfaces, which is invariant with respect to isometries. A set of reference points, evenly distributed on the point set surface, is sampled by farthest point sampling. The geodesic distance between reference points is normalized and stored in a geodesic distance matrix. Each row of the matrix yields a histogram of its elements. The set of histograms of the rows of a distance matrix is taken as a descriptor of the shape of the surface. The dissimilarity between two point set surfaces is computed by matching the corresponding sets of histograms with bipartite graph matching. This is an effective method for classifying and recognizing objects deformed with isometric transformations, e.g., non-rigid and articulated objects in different postures.

A type of rooted map called $(p,q,n)$-dipole, whose numbers on surfaces have some applications in string theory, are defined and the numbers of $(p,q,n)$-dipoles on orientable surfaces of genus 1 and 2 are given by Visentin and Wieler (The Electronic Journal of Combinatorics 14 (2007),#R12). In this paper, we study the classification of $(p,q,n)$-dipoles on nonorientable surfaces and obtain the numbers of $(p,q,n)$-dipoles on the projective plane and Klein bottle.

Let M be a non-orientable surface with Euler characteristic χ(M) ≤ −2. We consider the moduli space of flat SU(2)-connections, or equivalently the space of conjugacy classes of representations $$\mathfrak{X} (M) = {\rm Hom} (\pi_1 (M), {\rm SU} (2)) / {\rm SU} (2).$$There is a natural action of the mapping class group of M on \({\mathfrak{X} (M)}\). We show here that this action is ergodic with respect to a natural measure. This measure is defined using the push-forward measure associated to a map defined by the presentation of the surface group. This result is an extension of earlier results of Goldman for orientable surfaces (see [8]).

From Sphere to Torus: A Topological View of the Metazoan Body Plan

Let N_{g} denote a closed nonorientable surface of genus g. For g ge 2 the mapping class group mathcal {M}(N_{g}) is generated by Dehn twists and one crosscap slide (Y-homeomorphism) or by Dehn twists and a crosscap transposition. Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We give necessary and sufficient conditions for the existence of roots of crosscap slides and crosscap transpositions.

Asymptotic enumeration and limit laws for graphs of fixed genus

A map is a \(2\) -cell decomposition of a closed compact surface, i.e., an embedding of a graph such that every face is homeomorphic to an open disc. An automorphism of a map can be thought of as a permutation of the vertices, which preserves the vertex-edge-face incidences in the embedding. Every automorphism of a map determines an angle-preserving homeomorphism of the surface. While it is conjectured that there is no “truly subquadratic” algorithm for testing map isomorphism for unconstrained genus, we present a linear-time algorithm for computing the generators of the automorphism group of a map on an orientable surface of genus \(g\neq 0\) , parametrized by the genus \(g\) . A map on an orientable surface is uniform if the cyclic vector of sizes of faces incident to a vertex \(v\) does not depend on the choice of \(v\) . The algorithm applies a sequence of local reductions and produces a uniform map while preserving the automorphism group. The automorphism group of the original map can be reconstructed from the automorphism group of the associated uniform map in linear time. We also extend the algorithm to non-orientable surfaces by making use of the antipodal double-cover. The algorithm can be used to solve the map isomorphism problem between maps (orientable or non-orientable) of bounded negative Euler characteristic.