**Mark van Hoeij and Vijay Kunwar.**

**Definition 1:** Let **P ^{1} = C ⋃ {∞}** denote the Riemann sphere.
Let

- 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 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.

- 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**P**- {0, 1, ∞}. That gives an^{1}**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.