Last week, the great Naiomi Cameron visited me for a few days to discuss some new directions about Shabat polynomials. I’ve been horrible about posting on this blog, so now that I’ve been motivated to work on Shabat polynomials again, I figured it’s time for me to write!

As considered in the 1994 paper by Georgii Borisovich Shabat and Alexander Zvonkin entitled *Plane trees and algebraic numbers*, the rational function

is a Shabat polynomial which happens to be the composition

## Belyi Maps, Degree Sequences, and Passports

Let me begin by establishing some notation and making some remarks. A lot of this to begin with will be background information, so I apologize to those who already know this stuff pretty well.

Let

A Belyi map

Once we let

and the passport

where

These quantities give combinatorial information about the critical points of

## Shabat Polynomials

We say

we may as well assume that a Shabat polynomial

We often omit the information about infinity and write the degree sequence/passport of a Shabat polynomial as

where as before

There is a simple way to generate lots of examples of Shabat polynomials. Say that

## Dessins d’Enfants and Monodromy Groups

Given a Belyi map

- The “black” vertices are
; - The “white” vertices are
; - The “midpoints of the faces” are
; and - The “edges” are
as the inverse image of the line segment .

The Dessin d’Enfant will have

In addition to Belyi maps/Shabat polynomials

- Label the
edges of from 1 through . - For each “black” vertex
, read off the labels of the adjacent edges going counterclockwise, say as a -cycle in the symmetric group. - For each “white” vertex
, read off the labels of the adjacent edges going counterclockwise, say as a -cycle in the symmetric group. - For each “midpoint of edge”
, read off the labels of the adjacent edges going counterclockwise, say as a -cycle in the symmetric group. - Form that subgroup
of which is generated by the products of disjoint cycles , and .

If

It is not too difficult to recover the Dessin d’enfant

## Relating Belyi Maps and Monodromy Groups

Fix a positive integer

Proposition. The following are equivalent:

is the degree sequence of a Belyi map of degree on the Riemann sphere . - There is a triple
of elements in which can be expressed as a disjoint product of cycles , , and such that and is a transitive subgroup of .

The result was established by Adolf Hurwitz in his 1891 paper *Ueber Riemann’sche Flachen mit gegebenen Verzweigungspunkten*. I’ve explained how to construct a degree sequence from a Belyi map, but of course the opposite direction is an active area of research.

Proposition. All permutationsof cycle type are conjugate, that is, for some . In particular, all triples of “cycle type ” are conjugate, that is, , , and for some .

This is an elementary fact about permutations.

Proposition. There is a one-to-one correspondence between:

- Belyi maps
of degree on some compact connected Riemann surface . - Triples
of elements in , modulo simultaneous conjugation, such that and is a transitive subgroup of .

Proposition. Say thatis a triple of elements in such that , and denote as the subgroup of generated by them. Denote as the centralizer of in . There is a one-to-one correspondence between:

- The set of all triples
, modulo simultaneous conjugation, such that and , for some - Double cosets in
.

These correspondences between Belyi maps, triples *Numerical calculation of three-point branched covers of the projective line*. You can find the paper online at http://arxiv.org/abs/1311.2081v3. For the third correspondence, I’ve explained how to construct a triple of permutations from a Belyi map. Here’s a quick sketch of the fourth correspondence: say that

for some

in terms of

## Algorithm: Counting Belyi Maps from Degree Sequences

The monodromy group

- Fix a positive integer
. - Compute all multisets of multisets
such that

. - For each
, find a triple of elements in which can be expressed as a disjoint product of cycles , , and such that and is a transitive subgroup of . If no such triple exists, there is no Belyi map for which is a degree sequence. - Compute
and as the centralizers of and in . Given a double-coset representative in , the triple corresponding to and gives a unique Belyi map.

## Example #1

Let me give a relatively simple example. The rational

- the “black” vertex is
, - the “white” vertices are
for , - the “midpoint of the face” is
, and - the edges are the segments
for .

If we label edge

In fact, I can do even better: I claim this is the only Shabat polynomial associated with this passport! Indeed, if we have a triple

## Example #2

As considered in the 1994 paper by Georgii Borisovich Shabat and Alexander Zvonkin entitled *Plane trees and algebraic numbers*, the rational function

is a Shabat polynomial of degree

so that

so that

Question.is a composition of Shabat polynomials, where and . Since , do we have as the wreath product of by ?

I’m really confused by this example because it seems to be part of a general phenomenon… I will post more on this later as I come up with more examples — and possibly a proof!