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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a dessin d'enfant is a type of
graph embedding In topological graph theory, an embedding (also spelled imbedding) of a Graph (discrete mathematics), graph G on a surface (mathematics), surface \Sigma is a representation of G on \Sigma in which points of \Sigma are associated with graph the ...
used to study
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s and to provide combinatorial invariants for the action of the
absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' tha ...
of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s. 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 dessin d'enfant is a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
, with its vertices colored alternately black and white, embedded in an
oriented surface In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
that, in many cases, is simply a
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * Planes (gen ...
. For the coloring to exist, the graph must be bipartite. The faces of the embedding are required be topological disks. The surface and the embedding may be described combinatorially using a
rotation system In combinatorial mathematics, rotation systems (also called combinatorial embeddings or combinatorial maps) encode embeddings of graphs onto orientable surfaces by describing the circular ordering of a graph's edges around each vertex. A more for ...
, a
cyclic order In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "". One does not say that east is "more clockwise" than west. In ...
of the 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 it is embedded in with a structure as a Riemann surface. It is natural to ask which Riemann surfaces arise in this way. The answer is provided by
Belyi's theorem In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve ''C'', defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified a ...
, which states that the Riemann surfaces that can be described by dessins are precisely those that can be defined as
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s over the field of
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s. The absolute Galois group transforms these particular curves into each other, and thereby also transforms the underlying dessins. For a more detailed treatment of this subject, see or .


History


19th century

Early proto-forms of dessins d'enfants appeared as early as 1856 in the
icosian calculus The icosian calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856. In modern terms, he gave a group presentation of the icosahedral rotation group by generators and relations. Ham ...
of
William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
; in modern terms, these are
Hamiltonian path In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex ...
s on the icosahedral graph. Recognizable modern dessins d'enfants and
Belyi function In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve ''C'', defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at t ...
s were used by . Klein called these diagrams ''Linienzüge'' (German, plural of ''Linienzug'' "line-track", also used as a term for
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
); he used a white circle for the preimage of 0 and a '+' for the preimage of 1, rather than a black circle for 0 and white circle for 1 as in modern notation. He used these diagrams to construct an 11-fold cover of the Riemann sphere by itself, with
monodromy group In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...
PSL(2,11), following earlier constructions of a 7-fold cover with monodromy PSL(2,7) connected to the
Klein quartic In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms ...
in . These were all related to his investigations of the geometry of the quintic equation and the group collected in his famous 1884/88 ''Lectures on the Icosahedron''. The three surfaces constructed in this way from these three groups were much later shown to be closely related through the phenomenon of
trinity The Christian doctrine of the Trinity (, from 'threefold') is the central dogma concerning the nature of God in most Christian churches, which defines one God existing in three coequal, coeternal, consubstantial divine persons: God the F ...
.


20th century

Dessins d'enfant in their modern form were then rediscovered over a century later and named by Alexander Grothendieck in 1984 in his ''
Esquisse d'un Programme "Esquisse d'un Programme" (Sketch of a Programme) is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck in 1984. He pursued the sequence of logically linked ideas in his impor ...
''. quotes Grothendieck regarding his discovery of the Galois action on dessins d'enfants: Part of the theory had already been developed independently by some time before Grothendieck. They outline the correspondence between maps on topological 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. Later work by extends the treatment to surfaces with a boundary.


Riemann surfaces and Belyi pairs

The
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, together with a special point designated as \infty, form a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
known as the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
. Any
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
, and more generally any
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
p(x)/q(x) where p and q are polynomials, transforms the Riemann sphere by mapping it to itself. Consider, for example, the
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
f(x) = -\frac = 1 - \frac. At most points of the Riemann sphere, this transformation is a
local homeomorphism In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f : X \to Y is a local homeomorphism, X is said to be an é ...
: it maps a small disk centered at any point in a one-to-one way into another disk. However, at certain critical points, the mapping is more complicated, and maps a disk centered at the point in a k-to-one way onto its image. 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 Critical value may refer to: *In differential topology, a critical value of a differentiable function between differentiable manifolds is the image (value of) ƒ(''x'') in ''N'' of a critical point ''x'' in ''M''. *In statistical hypothesis ...
. The example given above, f, has the following critical points 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.) One may form a dessin d'enfant from f by placing black points at the
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
s of 0 (that is, at 1 and 9), white points at the preimages of 1 (that is, at 3\pm2\sqrt3), and arcs at the preimages of the
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
, 1 This line segment has four preimages, two along the line segment from 1 to 9 and two forming a
simple closed curve In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior ...
that loops from 1 to itself, surrounding 0; the resulting dessin is shown in the figure. In the other direction, from this dessin, described as a combinatorial object without specifying the locations of the critical points, one may form a
compact Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
, and a map from that surface to the Riemann sphere, equivalent to the map from which the dessin was originally constructed. To do so, place a point labeled \infty within each region of the dessin (shown as the red points in the second figure), and
triangulate In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
each region by connecting this point to the black and white points forming the boundary of the region, connecting multiple times to the same black or white point if it appears multiple times on the boundary of the region. Each triangle in the triangulation has three vertices labeled 0 (for the black points), 1 (for the white points), or \infty. For each triangle, substitute a half-plane, either the
upper half-plane In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to t ...
for a triangle that has 0, 1, and \infty in counterclockwise order or the lower half-plane for a triangle that has them in clockwise order, and for every adjacent pair of triangles glue the corresponding half-planes together along the portion of their boundaries indicated by the vertex labels. The resulting Riemann surface can be mapped to the Riemann sphere 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 In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic. Formal definit ...
. However, this construction identifies the Riemann surface only as a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
with complex structure; it does not construct an embedding of this manifold as an
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
in the
complex projective plane In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...
, although such an embedding always exists. The same construction applies more generally when X is any Riemann surface and f is a
Belyi function In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve ''C'', defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at t ...
; that is, a
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
f from X to the Riemann sphere having only 0, 1, and \infty as critical values. A pair (X,f) of this type is known as a ''Belyi pair''. From any Belyi pair (X,f) one can form a dessin d'enfant, drawn on the that has its black points at the preimages f^(0) of 0, its white points at the preimages f^(1) of 1, and its edges placed along the preimages f^( ,1 of the line segment ,1/math>. 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 halfspace by the identity to the Riemann sphere produces a Belyi function f on X, and therefore leads to a Belyi pair (X,f). Any two Belyi pairs (X,f) that lead to combinatorially equivalent dessins d'enfants are biholomorphic, and
Belyi's theorem In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve ''C'', defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified a ...
implies that, for any compact Riemann surface X defined over the
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s, there are a Belyi function f and a dessin d'enfant that provides a combinatorial description of both X


Maps and hypermaps

A vertex in a dessin has a graph-theoretic
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
, the number of incident edges, that equals its 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 ''clean'', and their corresponding Belyi functions are called ''pure''. When this happens, one can describe the dessin by a simpler embedded graph, one that has only the black points as its vertices and that has an edge for each white point with endpoints at the white point's two black neighbors. For instance, the dessin shown in the figure could be drawn more simply in this way as a pair of black points with an edge between them and a
self-loop In graph theory, a loop (also called a self-loop or a ''buckle'') is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow ...
on one of the points. 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. Thus, 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 as black points of a dessin, and placing white points at the midpoint of each embedded graph edge. If a map corresponds to a Belyi function f, its dual map (the dessin formed from the preimages of the line segment ,\infty/math>) corresponds to the 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 edges. The corresponding transformation of Belyi pairs is to replace a Belyi function \beta by the pure Belyi function \gamma=4\beta(1-\beta). One may calculate the critical points of \gamma directly from this formula: \gamma^(0)=\beta^(0)\cup\beta^(1), \gamma^(\infty)=\beta^(\infty), and \gamma^(1)=\beta^(\tfrac12). Thus, \gamma^(1) is the preimage under \beta of the midpoint of the line segment ,1/math>, and the edges of the dessin formed from \gamma subdivide the edges of the dessin formed from \beta. Under the interpretation of a clean dessin as a map, an arbitrary dessin is a ''hypermap'': that is, a drawing of a
hypergraph In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) wh ...
in which the black points represent vertices and the white points represent hyperedges.


Regular maps and triangle groups

The five
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s – the regular
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
,
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
,
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
,
dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
, and
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
– viewed as two-dimensional surfaces, have the property that any flag (a triple of a vertex, edge, and face that all meet each other) can be taken to any other flag by a symmetry of the surface. More generally, a map embedded in a surface with the same property, that any flag can be transformed to any other flag by a symmetry, is called a regular map. If a regular map is used 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 symmetry of the surface, and the reflections across those lines generate a symmetry group called a
triangle group In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle ...
, for which the triangles form the fundamental domains. 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 Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
is greater than one, the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of the surface is the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
, and the triangle group in the hyperbolic plane formed from the lifted triangulation is a (cocompact)
Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of t ...
representing a discrete set of isometries of the hyperbolic plane. In this case, the starting surface is the quotient of the hyperbolic plane by a finite
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
subgroup ''Γ'' in this group. Conversely, given a Riemann surface that is a quotient of a (2,3,n) tiling (a tiling of the sphere, Euclidean plane, or hyperbolic plane by triangles with angles \tfrac\pi2, \tfrac\pi3, and \tfrac\pi, the associated dessin is the
Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayle ...
given by the order two and order three generators of the group, or equivalently, the tiling of the same surface by n-gons meeting three per vertex. Vertices of this tiling give black dots of the dessin, centers of edges give white dots, and centers of faces give the points over infinity.


Trees and Shabat polynomials

The simplest bipartite graphs are the
trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
. Any embedding of a tree has a single region, and therefore by
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...
lies in a spherical surface. The corresponding Belyi pair forms a transformation of the Riemann sphere that, if one places the pole at \infty, can be represented as a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
. 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. The degree of the polynomial equals the number of edges in the corresponding tree. Such a polynomial Belyi function is known as a Shabat polynomial, after George Shabat. For example, take p to be the
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer exponent ...
p(x)=x^d having only one finite critical point and critical value, both
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
. Although 1 is not a critical value for p, it is still possible to interpret p as a Belyi function from the Riemann sphere to itself because its critical values all lie in the set \. The corresponding dessin d'enfant is a
star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
having one central black vertex connected to d white leaves (a
complete bipartite graph In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory ...
K_). More generally, a polynomial p(x) having two critical values y_1 and y_2 may be termed a Shabat polynomial. Such a polynomial may be normalized into a Belyi function, with its critical values at 0 and 1, by the formula q(x)=\frac, but it may be more convenient to leave p in its un-normalized form., p. 82. An important family of examples of Shabat polynomials are given by the
Chebyshev polynomials The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...
of the first kind, T_n(x), which have −1 and 1 as critical values. The corresponding dessins take the form of
path graph In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order such that the edges are where . Equivalently, a path with at least two vertices is connected and has two terminal ...
s, alternating between black and white vertices, with n edges in the path. Due to the connection between Shabat polynomials and Chebyshev polynomials, Shabat polynomials 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 absolute Galois group and its invariants

The polynomial p(x)=x^3(x^2-2x+a)^2 \, may be made into a
Shabat polynomial In mathematics, a 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 ...
by choosing a=\frac 7. 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. However, as these polynomials are defined over the
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
\mathbb(\sqrt), they may be transformed by the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of the
absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' tha ...
\Gamma of the rational numbers. An element of \Gamma that transforms \sqrt to -\sqrt will transform f_1 into f_2 and vice versa, and thus can also be said to transform each of the two trees shown in the figure into the other tree. More generally, due to the fact that the critical values of any Belyi function are the pure rationals 0, 1, and \infty, these critical values are unchanged by the Galois action, so this action takes Belyi pairs to other Belyi pairs. One may define an action of \Gamma on any dessin d'enfant by the corresponding action on Belyi pairs; this action, for instance,
permutes In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
the two trees shown in the figure. Due to Belyi's theorem, the action of \Gamma on dessins is faithful (that is, every two elements of \Gamma define different permutations on the set of dessins), so the study of dessins d'enfants can tell us much about \Gamma itself. In this light, it is of great interest to understand which dessins may be transformed into each other by the action of \Gamma and which may not. For instance, one may observe that the two trees shown have the same
degree sequence In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted ...
s 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 \Gamma transforms one dessin into another, both will have the same degree sequence. The degree sequence is one known
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
of the Galois action, but not the only invariant. The ''stabilizer'' of a dessin is the subgroup of \Gamma consisting of group elements that leave the dessin unchanged. Due to the Galois correspondence between subgroups of \Gamma and algebraic number 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 necessarily finite and stabilizers are of finite
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
. One may similarly define the stabilizer of an orbit (the subgroup that fixes all elements of the orbit) and the corresponding field of moduli of the orbit, another invariant of the dessin. The stabilizer of the orbit is the maximal
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of \Gamma contained in the stabilizer of the dessin, and the field of moduli of the orbit corresponds to the smallest normal extension of \mathbb that contains the field of moduli of the dessin. For instance, for the two conjugate dessins considered in this section, the field of moduli of the orbit is \mathbb(\sqrt). 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., pp. 122–123.


Notes


References

*. *. * . * *. Collected in . *. *. *. *. * , collected as pp. 140–165 i
Oeuvres, Tome 3
. * . See especially chapter 2, "Dessins d'Enfants", pp. 79–153. * . * . * . * . * . {{Algebraic curves navbox Complex analysis Algebraic geometry Topological graph theory