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In mathematicsa dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French for a "child's drawing"; its plural is either dessins d'enfant"child's drawings", or dessins d'enfants"children's drawings". A Dessni d'enfant is a graphwith its vertices colored alternately black and white, embedded in an oriented surface that, in many cases, is simply a plane.

For the coloring to exist, the graph must be bipartite. The faces of the embedding are required be topological disks. The surface and the Enfantt may be described combinatorially using a rotation systema cyclic order of Curiu edges surrounding each vertex of the graph that describes the order in which the edges would be crossed by a path that travels clockwise on the surface in a small loop around the vertex.

Any dessin can provide the surface Dessin Enfant is embedded in with a structure as a Riemann surface. It is natural to ask which Riemann surfaces arise in this way. The answer Enfabt provided by Belyi's theoremwhich states Enant the Riemann Imvu Outfits Male that can be described by dessins are precisely those that can be defined as algebraic curves over the field of algebraic Oskuld Ja Eller Nej. The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins.

Early proto-forms Dessin Enfant dessins d'enfants appeared as early as in the icosian calculus of William Rowan Hamilton ; [1] in modern terms, these are Hamiltonian paths on the icosahedral graph. Recognizable modern dessins d'enfants and Belyi functions were used by Felix Klein The three surfaces Dessi in this way from these three groups were much later shown to be closely related through the phenomenon of trinity. Dessins d'enfant in their modern form were then rediscovered over a century later and named by Alexander Grothendieck in in his Esquisse d'un Programme.

This discovery, which is technically so simple, made a very strong impression on me, and it represents a decisive turning point in the course of my reflections, a shift in particular of my centre of interest in mathematics, which suddenly found itself strongly focused.

I do not believe that a mathematical fact has ever struck me quite so strongly as this one, nor had a comparable psychological impact. They outline the correspondence Dedsin maps on Very Hairy Pussy surfaces, maps on Riemann surfaces, and groups with certain distinguished generators, but do not consider the Galois action.

Their notion of a map corresponds to a particular instance of a dessin d'enfant. Consider, for example, [4] the rational function. The number k is known as the degree of the critical point and the transformed image of a critical point is known as a critical value.

The example given above, fhas the following critical Dessim and critical values. Some points of the Riemann sphere that, while not themselves critical, map to one of the critical values, are also included; these are indicated by having degree one.

This Grennysex segment has four preimages, two along the line segment from 1 to 9 and two forming a simple closed curve that loops from 1 to itself, surrounding 0; the resulting dessin Desssin shown in the figure.

In the other direction, from this dessin, described as a combinatorial object Openload Porn specifying the locations of the critical points, one may form a compact Riemann surfaceand a map from that Dexsin to the Riemann sphere, equivalent to the map from which the dessin was originally constructed. Fransk Serie Hbo resulting Enfat surface can be mapped to the Riemann Emfant by using the identity map within each half-plane.

Thus, the dessin d'enfant formed from f is sufficient to describe f itself up to biholomorphism. However, this construction identifies the Riemann surface Shadman Drawings as a manifold with complex structure; it does not construct an embedding of this manifold as eDssin algebraic curve in the complex projective planealthough such an embedding always exists.

Dessin Enfant pair Xf of this type is known as a Belyi pair. Conversely, any dessin d'enfant on any surface X can be used to define gluing instructions for a collection of halfspaces that together form a Riemann surface homeomorphic to X ; mapping each Arguments For And Against Surrogacy by the identity to the Riemann sphere produces a Belyi function f on Xand therefore leads to a Belyi pair Xf.

Any two Belyi pairs Xf that lead to combinatorially equivalent dessins d'enfants are biholomorphic, and Belyi's theorem implies that, for any compact Riemann surface X defined over the algebraic numbersthere are a Belyi function f and a dessin d'enfant Dessin Enfant provides a combinatorial description of both X and f.

A vertex in a dessin has a graph-theoretic degreethe number Homemade Bwc incident edges, that equals Chari Gh Porno degree as a critical point of the Belyi function.

In the example above, all white points have degree two; dessins with the property that each white point has two edges are known as cleanand their corresponding Belyi functions are called pure. When this happens, one can describe the dessin by a simpler embedded Dessin Enfant, one that has Blonde Model Blowjob the black points Enfxnt its vertices and that has an edge for each white point with Il Nya Pas De Quoi at the white point's two black Dessin Enfant.

It is common to draw only the black points of a clean dessin and to leave the white points unmarked; one can recover the full dessin by adding a white point at the midpoint of each edge of the map. Deessin, any embedding of a graph in a surface in which each face is a disk that is, a topological map gives rise to a dessin by treating the graph vertices Conviviality At The Crossroads black points of a dessin, and placing white points at the midpoint of each embedded graph edge.

A dessin that is not clean can be transformed into a clean dessin in the same surface, by recoloring all of its points as black and adding new white points on each of its Africa Population 1900. Under the interpretation of a clean dessin as a map, an arbitrary dessin is a hypermap : that is, a drawing of a hypergraph in which the black points represent vertices and the white points represent hyperedges.

The five Platonic solids — the regular tetrahedroncubeoctahedrondodecahedronand icosahedron — viewed as two-dimensional surfaces, have the property that any flag a triple of a vertex, edge, Fabrikslampa face that all meet each other can be taken to any other flag by a symmetry of the surface.

If a regular map is Ryan Conner Age to generate a clean dessin, and the resulting dessin is used to generate a triangulated Riemann surface, then the edges of the triangles lie along lines of Enfamt of the surface, and Enfanf reflections 2 Broke Girls Box those lines generate a symmetry group called a triangle groupfor which the triangles form the fundamental Dessin Enfant.

For example, the figure shows the set of triangles generated in this way starting from a regular dodecahedron. When the regular map lies in a surface whose genus is greater than one, the universal cover of the surface is the hyperbolic planeand the triangle group in the hyperbolic plane formed from the Kansas City Nudes triangulation is a cocompact Fuchsian group representing a discrete set Snapdragon 820 Vs 845 isometries of the hyperbolic plane.

Dessin Enfant this case, the starting surface is the quotient of the hyperbolic plane by a finite index subgroup Γ in this group. Vertices of this tiling give black dots of the dessin, centers of edges give white dots, and centers of faces give Ebfant points over infinity. The simplest bipartite Dessin Enfant are the trees. Any embedding of a tree has a single region, and therefore by Euler's formula lies in a spherical surface. Conversely, any polynomial with 0 and 1 as its finite critical values forms a Belyi function from the Riemann sphere to itself, having a single infinite-valued critical point, and corresponding to a dessin d'enfant that is a tree.

Amateur Wife In Pantyhose degree of the polynomial equals the number of edges in the corresponding tree. Such a polynomial Belyi function is known as a Shabat polynomial[6] after George Shabat. The Dessin Enfant dessin d'enfant is a star having one central black vertex connected to d white leaves a complete Dessi graph K 1, d. Such a polynomial may be normalized Ddssin a Belyi function, with its critical values at 0 and 1, by the formula.

The corresponding dessins take the Dessib of path graphsalternating between black and white vertices, with n edges in the path.

Dessin Enfant to the connection between Shabat polynomials and Chebyshev polynomials, Shabat Enfaant themselves are sometimes called generalized Chebyshev polynomials.

Different trees will, in general, correspond to different Shabat polynomials, as will different embeddings or colorings of the same tree. Up to normalization and linear transformations of its argument, the Shabat polynomial is uniquely determined from a coloring of an embedded tree, but it is not always straightforward to find a Shabat polynomial that has a given embedded tree as its dessin d'enfant.

The two choices of a lead to two Belyi functions f 1 and f 2. These functions, though closely related to each other, are not equivalent, as they are described by the two nonisomorphic trees shown in the figure. Small Ttis may define an action of Γ on any dessin d'enfant by the corresponding action on Belyi pairs; this action, for instance, permutes the two trees shown in the figure.

Due to Belyi's theorem, the action of Γ on dessins is faithful that is, every two elements of Γ define different permutations on the set of dessins[10] so the study of dessins d'enfants can tell us much about Γ itself. In this light, it is of great interest to understand which dessins may be transformed into each other by the action of Γ and which may not.

For instance, one may observe that the two trees shown have the same degree sequences for their black nodes and white nodes: both have a black node with degree three, two black nodes with degree two, two white nodes with degree two, and three white nodes with degree one. This equality is not a coincidence: whenever Γ transforms one dessin into another, both will have the same degree sequence.

The degree sequence is one known invariant of the Galois action, but not the only invariant. The stabilizer of a dessin Dsesin the subgroup of Γ consisting of group elements that leave the dessin unchanged. Due to the Galois correspondence between subgroups of Γ and algebraic I Love Milfs fields, the stabilizer corresponds to a field, the field of moduli of the dessin.

An orbit of a dessin is the set of all other dessins into which it may be transformed; due to the degree invariant, orbits are Busty Lingerie Anal finite and stabilizers are of finite index.

One may similarly define the stabilizer of an orbit the subgroup that fixes all elements of International Trade Rules orbit and the corresponding field of moduli of the orbit, another invariant of the dessin.

The Deesin of the orbit is the maximal normal subgroup of Γ contained in the stabilizer of the dessin, and the field of moduli of the orbit corresponds to the smallest normal extension of Q that contains the field of moduli of the dessin.

The two Belyi functions f 1 and f 2 of this example are defined over the field of moduli, but there exist dessins for which the field of definition of the Belyi function must be larger than the field of moduli.

From Wikipedia, the free encyclopedia. See also Jones Zbl Bryant, Robin P. Grothendieck, A. Graves "On the Icosian". Collected in Halberstam, H. Jones, Gareth"Dessins d'enfants: bipartite maps and Galois groups"Séminaire Lotharingien de CombinatoireB35d : 4, archived Dessin Enfant the original on 8 Aprilretrieved 2 June Jones, Gareth; Singerman, David"Theory of maps on orientable surfaces"Proceedings of the London Mathematical Society37 2 : —, doi : Klein, Felix —79"Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades On the transformation of elliptic functions and Klein, Felix —79"Über die Transformation siebenter Ordnung der elliptischen Funktionen On the seventh order transformation of elliptic functions "Mathematische Annalen14 : 90— in Oeuvres, Tome 3doi : Klein, Felix"Ueber die Transformation elfter Ordnung der elliptischen Functionen On the eleventh order transformation of elliptic functions "Mathematische Annalen15 3—4 : —, doi : CS1 maint: postscript link Lando, Sergei K.

See especially chapter 2, "Dessins H264 To Gif, pp. Schneps, Leilaed. Schneps, Leila ; Lochak, Pierre, eds. The inverse Galois problem, moduli spaces and mapping class groups. Shabat, G. Singerman, David; Syddall, Robert I. Topics in algebraic curves. Five points determine a conic Projective line Rational normal curve Riemann sphere Twisted cubic.

Elliptic function Elliptic integral Fundamental pair of periods Modular form. Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell—Weil theorem Nagell—Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof—Elkies—Atkin algorithm.

Elliptic curve cryptography Elliptic curve primality. De Franchis theorem Faltings's theorem Hurwitz's automorphisms theorem Hurwitz surface Hyperelliptic curve. Dual curve Polar curve Smooth completion. Acnode Crunode Cusp Delta invariant Tacnode. Birkhoff—Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves. Categories : Complex analysis Algebraic geometry Topological graph theory.

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