A FIELD GUIDE TO DESSINS D’ENFANTS
A few black dots. A few white dots. Some lines.
And, hiding in plain sight, the symmetries of every algebraic number.
Children’s drawings. Graduate-level consequences.
A dessin d’enfant is a finite, connected bipartite graph drawn on a compact oriented surface, so that every region left over is a disc. The drawing is part of the data.
Colour the vertices black and white. Every edge must join opposite colours. Then remember the order in which the edges turn around each vertex. You may bend and stretch the picture; you may not casually swap that order. It is the difference between a graph and a map on a surface.
The extraordinary part is that this finite object determines a compact Riemann surface together with a special meromorphic function. Both admit algebraic equations over the algebraic numbers. An innocent drawing has become a piece of arithmetic geometry. [1]
An orientation-preserving homeomorphism of surfaces carrying one graph to the other and preserving the two colours. Vertex coordinates, edge curvature, and edge labels are not intrinsic. Reversing the orientation can change the dessin; exchanging the colours can too. Unless otherwise stated, all dessins here are connected.
Move between the pictures, the permutations, and the equations. Each viewpoint makes a different question easy.
Use finite sets with a transitive action of F₂ = ⟨a,b⟩ and equivariant maps; equivalently, connected finite coverings of U = ℙ¹(ℂ) ∖ {0,1,∞} and maps over U. Compactification gives Belyi pairs and holomorphic maps f with β′ ∘ f = β. On the dessin side, use the corresponding covering morphisms respecting colours and cyclic order—not arbitrary graph homomorphisms. Finite-index subgroups up to conjugacy classify isomorphism classes; retaining arrows requires the covering or F₂-set viewpoint. Passing to function fields reverses arrows, so that dictionary is an anti-equivalence. [1]
A Belyi map β : X → ℙ¹ is a nonconstant map whose branch values lie in {0, 1, ∞}. Pull back the interval [0,1]. There is your dessin.
For β(z) = zⁿ, the inverse image consists of n radial segments. The origin is one black vertex of valency n; the n roots of unity are white leaves. Infinity is the single pole, of order n. Only 0 and ∞ are branch values: “at most three” matters.
Locally, a ramification point looks like u ↦ uᵉ. Its ramification index e becomes a vertex valency over 0 or 1, or a face degree over ∞. The actual face boundary has 2e edge-sides, counted with repetitions.
BELYI’S THEOREMA smooth projective complex curve can be defined over Q̅ if and only if it admits a Belyi map.[1]
This is not a statement that every complex curve is arithmetic. It characterizes precisely those that are. Nor does a curve have just one dessin: the dessin determines the pair (X, β).
The same abstract graph. Three edges, two vertices. Change one cyclic order and the surface changes genus.
MONODROMY NOTEBOOK
Convention: permutations act on {1,2,3}; in σ₀σ₁, apply σ₁ first. Set σ∞ = (σ₀σ₁)⁻¹. Edge labels disappear under simultaneous conjugacy.
A cycle of σ₀ is a black vertex. A cycle of σ₁ is a white vertex. A cycle of σ∞ is a face. If c(σ) counts cycles, including fixed points, then
2 − 2g = c(σ₀) + c(σ₁) + c(σ∞) − n.
Thicken each vertex into an oriented disc and each edge into a ribbon, attached in the prescribed cyclic order. Cap each boundary component with a disc. You have reconstructed the surface. Transitivity of ⟨σ₀,σ₁⟩ is exactly connectedness.
Loops around 0 and 1 permute the sheets of an unbranched covering of the three-punctured sphere. The Riemann existence theorem supplies the corresponding analytic cover; fill in the punctures using the local maps u ↦ uᵉ. The result is a compact Riemann surface with a Belyi map. Its algebraic model exists over Q̅. Existence and uniqueness up to isomorphism do not hand you its coefficients—that is the computational adventure.
A compact record of how its sheets meet. A useful way to sort the travellers. Not a complete identity.
Here, the ramification passport means the ordered triple of partitions of n obtained from the cycle lengths of σ₀, σ₁, and σ∞. Other conventions include the genus and monodromy group. Always check what your database means. [2]
A passport gives the degree, the black and white valencies, the face degrees, and—by Euler’s formula—the genus. It does not generally determine a dessin, or even a single Galois orbit.
SPECIMEN / A FOUR-EDGE PATH
(2, 1, 1) | (2, 2) | (4)
β(z) = 1 − (z² − 1)²
σ₀ = (1 2)(3)(4), σ₁ = (1 3)(2 4)
For β(z) = 1 − (z² − 1)², the black roots are −√2, 0, √2 with multiplicities 1,2,1; the white roots are ±1, each with multiplicity 2. Its sole pole has order 4. The interval [−√2,√2] is the dessin, with alternating black and white vertices. The cyclic orders can be encoded by the permutation pair above.
Not competing definitions: overlapping families, each exposing a different kind of structure.
A tree embedded in the oriented sphere has one face. Its Belyi map has one pole; put that pole at infinity and the map becomes a polynomial, often called a Shabat polynomial. Even plane trees support a faithful absolute Galois action. Simplicity does not mean arithmetic innocence.
Every white vertex has valency 2. Suppress the white vertices and recover an ordinary embedded graph. Conversely, subdivide each edge of a map to make it clean. From a Belyi map β, the composition 4β(1−β) produces a clean dessin.
The automorphism group acts transitively on edges; the corresponding covering is Galois. Equivalently, its subgroup of F₂ is normal. The star for zⁿ is a first example: multiplication by an nth root of unity rotates its edges.
Triangulations arise by pulling back a triangulation of the sphere with vertices 0,1,∞. Hypermaps package the same permutation data. Triangle groups enter when ramification orders are bounded by a prescribed triple; their geometry may be spherical, Euclidean, or hyperbolic. Modular curves provide particularly rich arithmetic examples. These doors lead well beyond planar trees. [1] [2]
Gal(Q̅/ℚ) acts on algebraic coefficients. Through Belyi’s dictionary, that action reaches all the way back to finite drawings.
Every nonidentity element of the absolute Galois group moves some dessin. Astonishingly, plane trees already suffice to detect every such element. This is a theorem about the whole collection, not a claim that every individual dessin moves. Our examples with equations over ℚ are fixed. [1]
Degree, genus, ramification passport, and the isomorphism type of the monodromy group are Galois invariants. These constraints are powerful. They still do not give a complete practical test for belonging to the same Galois orbit.
Reflect the oriented surface. Complex conjugation reverses the cyclic orders, giving the mirror dessin. Some dessins are equivalent to their mirrors; others are not. Do not confuse a colour swap with a mirror reflection.
The field of moduli is the fixed field of the subgroup of Gal(Q̅/ℚ) stabilizing the isomorphism class of the Belyi pair. A field of definition is a field over which equations for that pair exist. Descent can be obstructed: the two notions need not coincide. Regular dessins do descend to their fields of moduli. Even there, surprising fields occur: examples of regular dessins of genus 61 have field of moduli ℚ(∛2). [3]
Which drawings are arithmetic siblings?
We can enumerate finite combinatorics. Recognizing its Galois orbits is another matter.Existence is not an algorithm you can run. And a convincing numerical answer is not yet an exact one.
Choose a transitive permutation pair. Compute cycles, genus, and monodromy. Simultaneous relabelling changes the encoding, not the dessin.
Translate prescribed ramification into polynomial conditions, or use analytic approximation, modular forms, and p-adic lifting. Symmetries help reduce the search.
Recover algebraic coefficients from sufficiently precise approximations. Precision, coefficient heights, and number-field degrees can make a small drawing expensive.
Verify exact ramification and compute monodromy. A matching passport is insufficient: equations can describe the wrong member of the same passport.
Take β(z) = 1 − (z² − 1)². Find its critical points, their images, and its pole order. Can you recover the passport without drawing it?
β′(z) = −4z(z² − 1). The critical points are 0 and ±1. Their images are 0 and 1 respectively. The fibre above 0 is {−√2,0,√2}, with multiplicities 1,2,1. The fibre above 1 is {−1,1}, with multiplicities 2,2. Infinity is a pole of order 4. Thus the passport is ((2,1,1),(2,2),(4)); the ramification contributions total 1+2+3 = 6 = 2·4−2, as Riemann–Hurwitz requires for a degree-four map between spheres.
Belyi maps give concrete equations for covers and arithmetic curves. They turn abstract existence questions into objects we can calculate, compare, and put in databases. Monodromy links them to inverse Galois constructions; descent and the branch locus still matter.
Explore the computational survey ↗Melanie Wood used Belyi-extending maps to construct further Galois invariants, distinguishing examples that the other readily computable invariants considered in that work did not separate. New ways of looking reveal arithmetic distinctions concealed by the original picture.
Read about extending invariants ↗Guillemot and Voight describe rigorous monodromy computation from exact number-field equations using certified path tracking, and apply it to Belyi maps in the LMFDB. The return trip—from equations to the intended dessin—is a mathematical result in its own right.
Read the 2026 preprint ↗Grothendieck’s vision places these finite objects within the study of fundamental groups and the absolute Galois group. The profinite completion of F₂ remembers all finite covers at once. The Galois action on this tower leads toward Grothendieck–Teichmüller theory; it is not a solved combinatorial description of Gal(Q̅/ℚ).
Follow the group-theoretic route ↗Read in layers: first the dictionary, then the algorithms, then the arithmetic that refuses to fit neatly inside either.
The main mathematical backbone: equivalences, monodromy, Galois actions, and the route to profinite groups.
A survey of methods, examples, applications, and the obstacles between a dessin and exact equations.
Galois orbits, descent, and regular dessins with non-abelian fields of moduli.
Constructing finer invariants by composing with suitable Belyi maps.
2026 preprint: rigorous identification of monodromy from exact equations.
Leave the atlas and visit actual records. Compare passports, equations, and monodromy data.
THE DESSIN ATLAS