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As I understand it, rooted maps on surfaces were first introduced in enumerative combinatorics because they are easier to count than unrooted maps, which can have non-trivial symmetries. A map is a graph $G$ embedded on a surface $X$ (such that every face in $X \setminus G$ is homeomorphic to a disc), while a rooted map is essentially a map together with a choice of a dart, i.e., an edge together with an orientation of that edge. (The "essentially" is because for counting purposes, it is also convenient to consider the vertex map---with one vertex and no edges---to be rooted by convention.) Rooted maps are always considered up to root-preserving homeomorphism. A similar notion of rooting extends to hypermaps, which are maps equipped with a two-coloring of vertices.

I'm interested in whether the idea of rooting has any independent motivation for dessin d'enfants, i.e., when viewing a hypermap as a representation of a Belyi function $f : X \to \bar{\mathbb{C}}$. It seems that a "rooted Belyi function" would correspond to a triple $(X,f,\gamma)$, where $\gamma : [0,1] \to X$ is a path containing no other critical points of $f$ besides $\gamma_0$ and $\gamma_1$ and such that $f(\gamma) = [0,1]$. (Is there a better way of putting that?)

Is such a concept natural from the point of view of algebraic geometry, or has something like it already been studied?

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For dessin d'enfants, I believe the orientation is superfluous - each edge goes between a black vertex and a white vertex, so picking an orientation is just picking one of those, whcih doesn't help uniformize anything.

A dessin d'enfants corresponds to a cover of $\mathbb P^1$ ramified over three points $(0,1,\infty)$. A rooted dessin d'enfants, I think, should be a dessin d'enfants with a marked edge. (Note that the trivial dessin has an edge, so no additional complexities are needed.) It's pretty easy to see that this corresponds to, after fixing a point $\lambda \not\in \{0,1,\infty\}$ in $\mathbb P^1$, taking a ramified cover and fixing a point lying over $\lambda$.

So that sounds pretty natural to me.

This is closely related to some algebraic geometry concepts (curve with marked points, monodromy of a cover), but I'm not sure if this exact concept has been studied, nor do I know whether studying this exact concept will yield new insights.

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  • $\begingroup$ Thanks! Your simple formulation of rooted dessins d'enfants makes sense to me, and I appreciate the pointers. $\endgroup$ Oct 4 '14 at 12:48
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Summary

For me, the idea of rooting has an important independent motivation for dessin d'enfant (which was the original question). I conjecture that it allows us to construct from each dessin $D$ an algebraic number $\eta_D$, in a systematic and useful way. I've started some computer calculations to investigate this question.

Rooted trees and bias

In Bias and Dessins (https://arxiv.org/pdf/1506.06389.pdf), Definition 2.11 (page 7) I wrote:

A biased Shabat polynomial is a polynomial function $f:\mathbb{C}\to\mathbb{C}$ such that (i) if $f'(u)=0$ then $f(u)\in\{-1, +1\}$, (ii) $f(0)=0$, and (iii) $f'(0)=1$.

For me, a biased $X$ is an $X$ with additional information added (in a simple and uniform manner) to give something that has no automorphisms. Rooting, as in Noam Zeilenberger's original post, is the sensible way to bias dessins. It requires no special knowledge of the dessin.

A surprising fact

In Bias and Dessins I show that every biased Shabat polynomial has associated to it many algebraic numbers, namely the coefficients of the polynomial function $f$. (I see no simple way to usefully combine these numbers into a single algebraic number.)

Going the other way, dessins to algebraic number, surprised me. (It's fairly obvious if you know some algebraic geometry over number fields. Bias and Dessins arose from a conference where most of the participants were combinatorialists. Hence the laboured proof, relying on properties of Shabat polynomials.)

From dessin $D$ construct algebraic number $\eta_D$

Subsequently, in The algebra of balanced dessins (https://arxiv.org/pdf/1802.04531.pdf) I wrote:

This paper gives a key definition, for a new approach to dessins and algebraic numbers. The distant goal is to construct from each dessin $D$ an algebraic number $\eta_D$, in a systematic and useful way.

This paper, which is only 2 pages long, gives key concepts for the definition of $\eta_D$. It also has a link to some calculations (https://github.com/jfine2358/pydessins), which I hope to pick up again this year.

Aside

Perhaps, applied to biased Shabat polynomials, this gives a useful way of constructing from $f$ a single algebraic number. But I don't see a sensible way to go from the conjectural $\eta_D$ to an $\eta_D$ lying in $\mathbb{C}$. The coefficients of a biased Shabat polynomial are not only algebraic numbers, they are algebraic numbers lying in $\mathbb{C}$. (For me, this points to a subtle aspect of $\eta_D$. I hope to write up something on this later this year.)

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