Enumeration of simple bipartite maps on the sphere and the projective plane

AbstractThis paper is concerned with the number of rooted simple bipartite maps on the plane according to the root-face valencies and the number of edges of the maps. An explicit formula with one parameter is given. Furthermore, a special kind of rooted simple bipartite maps on the projective plane are counted and, as special cases, recursive formulae for maps with fewer faces on the projective plane are presented as well

A bijection for nonorientable general maps

We give a different presentation of a recent bijection due to Chapuy and Dol\k{e}ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier--Di Francesco--Guitter-like generalization of the Cori--Vauquelin--Schaeffer bijection in the context of general nonorientable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and this allows us to recover a famous asymptotic enumeration formula found by Gao

Basic nets in the projective plane

The notion of basic net (called also basic polyhedron) on $S^2$ plays a central role in Conway's approach to enumeration of knots and links in $S^3$. Drobotukhina applied this approach for links in $RP^3$ using basic nets on $RP^2$. By a result of Nakamoto, all basic nets on $S^2$ can be obtained from a very explicit family of minimal basic nets (the nets $(2\times n)^*$, $n\ge3$, in Conway's notation) by two local transformations. We prove a similar result for basic nets in $RP^2$. We prove also that a graph on $RP^2$ is uniquely determined by its pull-back on $S^3$ (the proof is based on Lefschetz fix point theorem).Comment: 14 pages, 15 figure

A bijection for rooted maps on general surfaces

We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, that is orientable and non-orientable as well. This general construction requires new ideas and is more delicate than the special orientable case, but it carries the same information. In particular, it leads to a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable rooted connected maps of Euler characteristic $2-2h$, and of the algebraicity of their generating functions, similar to the one previously obtained in the orientable case via the Marcus-Schaeffer bijection. It also shows that the renormalization factor $n^{1/4}$ for distances between vertices is universal for maps on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation on any fixed surface converge in distribution when the size $n$ tends to infinity. Finally, we extend the Miermont and Ambj{\o}rn-Budd bijections to the general setting of all surfaces. Our construction opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.Comment: v2: 55 pages, 22 figure

A bijection for rooted maps on general surfaces (extended abstract)

International audienceWe extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable maps of Euler characteristic $2-2h$ and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor $n$¼ for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size $n$ on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.Nous étendons la bijection de Marcus et Schaeffer entre quadrangulations biparties orientables (de manière équivalente: cartes enracinées) et cartes à une face étiquetées orientables à toutes les surfaces, orientables ou non. Cette construction générale requiert des idées nouvelles et est plus délicate que dans le cas particulier orientable, mais permet des utilisations similaires. Elle donne donc une interprétation combinatoire uniforme de l’exposant de comptage $\frac{5(h-1)}{2}$ pour les cartes orientables et non-orientables de caractéristique d’Euler $2-2h$, et de l’algébricité des fonctions génératrices. Elle montre l’universalité du facteur de normalisation $n$¼ pour la métrique des cartes, sur toutes les surfaces: le profil et le rayon d’une quadrangulation enracinée pointée sur une surface fixée converge en distribution. Enfin, elle ouvre à la voie à l’étude des surfaces Browniennes pour toute 2-variété compacte

Feynman Diagrams and Rooted Maps

The Rooted Maps Theory, a branch of the Theory of Homology, is shown to be a powerful tool for investigating the topological properties of Feynman diagrams, related to the single particle propagator in the quantum many-body systems. The numerical correspondence between the number of this class of Feynman diagrams as a function of perturbative order and the number of rooted maps as a function of the number of edges is studied. A graphical procedure to associate Feynman diagrams and rooted maps is then stated. Finally, starting from rooted maps principles, an original definition of the genus of a Feynman diagram, which totally differs from the usual one, is given.Comment: 20 pages, 30 figures, 3 table

Problems on Polytopes, Their Groups, and Realizations

The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete Geometry, to appear

Quantum contextual finite geometries from dessins d'enfants

We point out an explicit connection between graphs drawn on compact Riemann surfaces defined over the field $\bar{\mathbb{Q}}$ of algebraic numbers --- so-called Grothendieck's {\it dessins d'enfants} --- and a wealth of distinguished point-line configurations. These include simplices, cross-polytopes, several notable projective configurations, a number of multipartite graphs and some 'exotic' geometries. Among them, remarkably, we find not only those underlying Mermin's magic square and magic pentagram, but also those related to the geometry of two- and three-qubit Pauli groups. Of particular interest is the occurrence of all the three types of slim generalized quadrangles, namely GQ(2,1), GQ(2,2) and GQ(2,4), and a couple of closely related graphs, namely the Schl\"{a}fli and Clebsch ones. These findings seem to indicate that {\it dessins d'enfants} may provide us with a new powerful tool for gaining deeper insight into the nature of finite-dimensional Hilbert spaces and their associated groups, with a special emphasis on contextuality.Comment: 18 page