Table of (almost)-Belyi maps with Five Exceptional Points

Mark van Hoeij and Vijay Kunwar.

Definition 1: Let P¹ = C ⋃ {∞} denote the Riemann sphere. Let k,l,m be positive integers (or ∞, see footnote 1) and let f: P¹ → P¹ be a rational function. The (k,l,m)-exceptional points of f are:

roots of f of order not divisible by k,
roots of 1-f of order not divisible by l,
roots of 1/f of order not divisible by m.

Footnote 1: If k = ∞ then all roots of f are (k,l,m)-exceptional points (and likewise for roots of 1-f resp. 1/f if l resp. m is ∞).

Footnote 2: Definition 1 differs slightly from Hoeij+Vidunas which defined exceptional points as
{roots of f of order ≠ k} ⋃ {roots of 1-f of order ≠ l} ⋃ {roots of 1/f of order ≠ m}.
In the files on this website, count=5 refers to definition 1, while Count=5 refers to the definition from Hoeij+Vidunas. The Count=5 table differs from the count=5 table below in two ways: The Belyi⁽ⁱ⁾ ∈ Belyi⁽ⁱ⁺¹⁾ cases disappear (explanation) and there are 5 and 1 additional Belyi maps in the first two entries of the first column.

Goal: (see section "Motivation") Up to Mobius-equivalence, list every rational function f with 5 (k,l,m)-exceptional points, with (k,l,m) = (∞,2,m) and m ∈ {3,4,6}.

count=5
(k,l,m)	Belyi ∉ Belyi⁽¹⁾	Belyi ∈ Belyi⁽¹⁾ indirectly + directly	Belyi⁽¹⁾ ∉ Belyi⁽²⁾	Belyi⁽¹⁾ ∈ Belyi⁽²⁾	Belyi⁽²⁾
(∞,2,3)	411 maps	9 + 266 maps	65 families	3 families	2 families
(∞,2,4)	121 maps	3 + 23 maps	20 families	no cases	no cases
(∞,2,6)	54 maps	2 + 5 maps	12 families	no cases	no cases

Completeness: Proving that the above table is complete is the goal of the paper and the accompanying website.

Belyi maps: Functions whose branched set is ⊆ {0, 1, ∞}.

Belyi⁽¹⁾ resp. Belyi⁽²⁾: These columns list functions with 1 resp. 2 branchpoints outside of {0, 1, ∞}. Since these branchpoint(s) are allowed to vary, these functions appear in 1 resp. 2 dimensional families.

Belyi⁽¹⁾ maps: These are the highlight of the table because

Combined with the Heun table they contain many of the Belyi maps (see column Belyi ∈ Belyi⁽¹⁾).
It is not a priori obvious that all of the Belyi⁽¹⁾ families should allow "perfect parametrizations" meaning parametrizations that:
- are rational
- cover, by direct substitution, all Belyi⁽¹⁾ functions with count=5 up to Mobius-equivalence, without any gaps
- and without Mobius-duplicates.
Details in items 3-5 below.
Our Belyi⁽¹⁾ families can be rationally parametrized. We use the variable s to parametrize these families, so each of our 1-dimensional families of Belyi⁽¹⁾ maps in C(x) is represented by a single f ∈ Q(s)(x). Values of s where f gives a Belyi map, with the same x-degree as f, are listed in the column Belyi ∈ Belyi⁽¹⁾.
Every Belyi⁽¹⁾ function with count=5 is Mobius-equivalent to precisely one f in our Belyi⁽¹⁾ table, for precisely one value of s.
For a Belyi⁽¹⁾ function f, that one branchpoint outside of {0, 1, ∞}, lets call it t, can be moved through P¹ - {0, 1, ∞}. That gives an action of the braid group on the set of all Belyi⁽¹⁾ maps that ramify above {0, 1, t, ∞}. For each f ∈ Q(s)(x) in our tables the following holds: Writing t in terms of s gives a Belyi map t ∈ Q(s), f covers precisely one braid orbit O which (up to Mobius-equivalence) has precisely |O| = deg_s(t) elements (key to prove uniqueness for the value of s).

count > 5: Computing all Belyi⁽¹⁾ maps is problematic for count=6. For Belyi maps, here are all dessins with Count ≤ 6. You can obtain the dessins for count ≤ 7 or Count ≤ 7 by downloading our algorithm in section 4.4.