scholarly journals Counting unicellular maps on non-orientable surfaces

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Olivier Bernardi ◽  
Guillaume Chapuy
Keyword(s):  
Connected Graph ◽  
Topological Type ◽  
Orientable Surface ◽  
Recurrence Equation ◽  
Asymptotic Formulas ◽  
Topological Disk ◽  
International Audience ◽  
Fixed Topology ◽  
Orientable Surfaces

International audience A unicellular map is the embedding of a connected graph in a surface in such a way that the complement of the graph is a topological disk. In this paper we give a bijective operation that relates unicellular maps on a non-orientable surface to unicellular maps of a lower topological type, with distinguished vertices. From that we obtain a recurrence equation that leads to (new) explicit counting formulas for non-orientable precubic (all vertices of degree 1 or 3) unicellular maps of fixed topology. We also determine asymptotic formulas for the number of all unicellular maps of fixed topology, when the number of edges goes to infinity. Our strategy is inspired by recent results obtained for the orientable case [Chapuy, PTRF 2010], but significant novelties are introduced: in particular we construct an involution which, in some sense, ``averages'' the effects of non-orientability. \par Une carte unicellulaire est le plongement d'un graphe connexe dans une surface, tel que le complémentaire du graphe est un disque topologique. On décrit une opération bijective qui relie les cartes unicellulaires sur une surface non-orientable aux cartes unicellulaires de type topologique inférieur, avec des sommets marqués. On en déduit une récurrence qui conduit à de (nouvelles) formules closes d'énumération pour les cartes unicellulaires précubiques (sommets de degré 1 ou 3) de topologie fixée. On obtient aussi des formules asymptotiques pour le nombre total de cartes unicellulaires de topologie fixée, quand le nombre d'arêtes tend vers l'infini. Notre stratégie est motivée par de récents résultats dans le cas orientable [Chapuy, PTRF, 2010], mais d'importantes nouveautés sont introduites: en particulier, on construit une involution qui, en un certain sens, "moyenne'' les effets de la non-orientabilité.

EXHAUSTION OF THE CURVE GRAPH VIA RIGID EXPANSIONS

2018 ◽  
Vol 61 (1) ◽  
pp. 195-230 ◽  
Author(s):  
JESÚS HERNÁNDEZ HERNÁNDEZ
Keyword(s):  
Topological Type ◽  
Orientable Surface ◽  
Finite Topological Type ◽  
Curve Graph ◽  
Finite Set ◽  
Rigid Set

AbstractFor an orientable surfaceSof finite topological type with genusg≥ 3, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph$\mathcal{C}$(S). The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set in Aramayona and Leininger,J. Topology Anal.5(2) (2013), 183–203 and Aramayona and Leininger,Pac. J. Math.282(2) (2016), 257–283, and in fact a consequence of our proof is that Aramayona and Leininger's set also exhausts the curve graph via rigid expansions.

On the homeotopy group of a non-orientable surface

1972 ◽  
Vol 71 (3) ◽  
pp. 437-448 ◽  
Author(s):  
Joan S. Birman ◽  
D. R. J. Chillingworth
Keyword(s):  
Projective Planes ◽  
Double Cover ◽  
Orientable Surface ◽  
Factor Group ◽  
Simple Type ◽  
Defining Relations ◽  
Entire Structure ◽  
Finite Set ◽  
Orientable Surfaces ◽  
Made In

Let X be a closed, compact connected 2-manifold (a surface), which we will denote by O or N if we wish to stress that X is orientable or non-orientable. Let G(X) denote the group of all homeomorphisms X → X, D(X) the normal subgroup of homeomorphisms isotopic to the identity, and H(X) the factor group G(X)/D(X), i.e. the homeotopy group of X. The problem of determining generators for H(O) was considered by Lickorish in (7, 8), and the second of these papers specifies a finite set of generators of a particularly simple type. In (10) and (11) Lickorish considered the analogous problem for non-orientable surfaces, and, using Lickorish's partial results, Chilling-worth (4) determined a finite set of generators for H(N). While the generators obtained for H(O) and H(N) were strikingly similar, it was noteworthy that the techniques used in the two cases were different, and in particular that little use was made in the non-orientable case of the earlier results obtained on the orientable case. The purpose of this note is to show that the results of Lickorish and Chillingworth on non-orientable surfaces follow rather easily from the work in (7, 8) by an application of some ideas from the theory of covering spaces (2). Moreover, while Lickorish and Chillingworth sought only to find generators, we are able to show (Theorem 1) how in fact the entire structure of the group H(N) is determined by H(O), where O is an orientable double cover of N. Finally, we are able to determine defining relations for H(N) for the case where N is the connected sum of 3 projective planes (Theorem 3).

scholarly journals The First Integral Cohomology of Pure Mapping Class Groups

2020 ◽  
Author(s):  
Javier Aramayona ◽  
Priyam Patel ◽  
Nicholas G Vlamis
Keyword(s):  
Cohomology Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
First Integral ◽  
Orientable Surface ◽  
Mapping Class ◽  
Class Groups ◽  
Simplicial Homology ◽  
Integral Cohomology ◽  
Orientable Surfaces

Abstract It is a classical result that pure mapping class groups of connected, orientable surfaces of finite type and genus at least 3 are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface’s simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.

On Non-Integral Dehn Surgeries Creating Non-Orientable Surfaces

2006 ◽  
Vol 49 (4) ◽  
pp. 624-627
Author(s):  
Masakazu Teragaito
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Orientable Surface ◽  
Dehn Surgery ◽  
Orientable Surfaces

AbstractFor a non-trivial knot in the 3-sphere, only integral Dehn surgery can create a closed 3-manifold containing a projective plane. If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true. In contrast to these, we show that non-integral surgery on a hyperbolic knot can create a closed non-orientable surface of any genus greater than two.

scholarly journals Some results on Teichmüller spaces of Klein surfaces

1997 ◽  
Vol 39 (1) ◽  
pp. 65-76
Author(s):  
Pablo Arés Gastesi
Keyword(s):  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Topological Type ◽  
Uniformization Theorem ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces ◽  
Simply Connected Domain ◽  
Nonorientable Surfaces ◽  
Simply Connected ◽  
Orientable Surfaces

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.

UNKNOTTING ORIENTABLE SURFACES IN THE 4-SPHERE

1995 ◽  
Vol 04 (02) ◽  
pp. 213-224 ◽  
Author(s):  
JONATHAN A. HILLMAN ◽  
AKIO KAWAUCHI
Keyword(s):  
Fundamental Group ◽  
Orientable Surface ◽  
Closed Orientable Surface ◽  
Flat Embedding ◽  
Orientable Surfaces

We show that a topologically locally flat embedding of a closed orientable surface in the 4-sphere is isotopic to one whose image lies in the equatorial 3-sphere if and only if its exterior has an infinite cyclic fundamental group.

scholarly journals On the self-similarity problem for smooth flows on orientable surfaces

2011 ◽  
Vol 32 (5) ◽  
pp. 1615-1660 ◽  
Author(s):  
JOANNA KUŁAGA
Keyword(s):  
Orientable Surface ◽  
The Self ◽  
Self Similarity ◽  
Similarity Problem ◽  
Orientable Surfaces

AbstractOn each compact connected orientable surface of genus greater than one we construct a class of flows without self-similarities.

scholarly journals Cubic Graphs and Related Triangulations on Orientable Surfaces

10.37236/5989 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Wenjie Fang ◽  
Mihyun Kang ◽  
Michael Moßhammer ◽  
Philipp Sprüssel
Keyword(s):  
Analogous Result ◽  
Orientable Surface ◽  
Negative Integer ◽  
Cubic Graphs ◽  
Simple Cubic ◽  
Planar Component ◽  
Orientable Surfaces

Let $\mathbb{S}_g$ be the orientable surface of genus $g$ for a fixed non-negative integer $g$. We show that the number of vertex-labelled cubic multigraphs embeddable on $\mathbb{S}_g$ with $2n$ vertices is asymptotically $c_g n^{5/2(g-1)-1}\gamma^{2n}(2n)!$, where $\gamma$ is an algebraic constant and $c_g$ is a constant depending only on the genus $g$. We also derive an analogous result for simple cubic graphs and weighted cubic multigraphs. Additionally, for $g\ge1$, we prove that a typical cubic multigraph embeddable on $\mathbb{S}_g$ has exactly one non-planar component.

scholarly journals The Choice Number Versus the Chromatic Number for Graphs Embeddable on Orientable Surfaces

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Brahadeesh Sankarnarayanan ◽  
Niranjan Balachandran
Keyword(s):  
Chromatic Number ◽  
Orientable Surface ◽  
Choice Number ◽  
Orientable Surfaces

We show that for loopless $6$-regular triangulations on the torus the gap between the choice number and chromatic number is at most $2$. We also show that the largest gap for graphs embeddable in an orientable surface of genus $g$ is of the order $\Theta(\sqrt{g})$, and moreover for graphs with chromatic number of the order $o(\sqrt{g}/\log_{2}(g))$ the largest gap is of the order $o(\sqrt{g})$.

scholarly journals Cyclic sieving phenomenon in non-crossing connected graphs

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Alan Guo
Keyword(s):  
Generating Function ◽  
Connected Graph ◽  
Connected Graphs ◽  
Cyclic Sieving Phenomenon ◽  
International Audience ◽  
Cyclic Sieving

International audience A non-crossing connected graph is a connected graph on vertices arranged in a circle such that its edges do not cross. The count for such graphs can be made naturally into a q-binomial generating function. We prove that this generating function exhibits the cyclic sieving phenomenon, as conjectured by S.-P. Eu. Un graphe connexe dont les sommets sont disposés sur un cercle est sans croisement si ses arêtes ne se croisent pas. Nous démontrons une conjecture de S.-P. Eu affirmant que la fonction génératrice q-binomiale dénombrant de tels graphes exhibe le phénomène du crible cyclique.

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