# Acknowledgements

We are grateful to Jérémie Bouttier, Séverin Charbonnier, Bertrand Duplantier, Emmanuel Guitter, Paul Norbury, and Paul Roux, for valuable discussions and correspondence. Moreover, we wish to thank Jérémie Bouttier, Emmanuel Guitter, Adam Nahum and Paul Roux for helpful suggestions on the draft text. We are grateful to Bernard Nienhuis for the review he wrote for SciPost, which led to significant clarifications.

Rongvoram Nivesvivat gratefully acknowledges the personal hospitality of the Jia family in Beijing in Fall 2022.

**Funding information** This work is partly a result of the project ReNewQuantum, which received funding from the European Research Council. This work was also supported by the French Agence Nationale de la Recherche (ANR) under grant ANR-21-CE40-0003 (project CONFICA).

# A List of examples

In this appendix we systematically display planar maps by increasing number of edges $\sum r_i \in \mathbb{N}$. We display all maps with $\sum r_i \le 3$, and all connected maps with $\sum r_i \le 5$.

To save space and avoid redundancies, we sometimes make use of the symmetry of our sets of maps under permutations of vertices and/or edges. Such symmetries exist whenever two or more vertices have the same valency. In such cases, a displayed map may come with a multiplicity $m$, which is the size of a relevant group of permutations. (We do not necessarily use the largest available group.) Then $m-1$ is the number of similar maps that we do not display explicitly.

For each map, we indicate the signature $\sigma$, see Definition 2.6. Two maps that are related by a symmetry may or may not have the same signature. The signature is written in **bold** if our set of maps $M_{0,4}$ contains no other map with a larger or equal signature $\sigma' \ge \sigma$ — in other words, if $|M_{0,4}(\delta_i|\sigma)| = 1$. In this case, Conjecture 4.4 states that the corresponding crossing symmetry equations have a one-dimensional space of solutions.

## A.1 All planar maps with zero to three edges

$$
\begin{array}{l}
\text{A.1.1 Case } |M_{0,4}(0,0,0,0)| = 1 \\
\qquad \times \times \\
\qquad \times \times \\
\qquad \sigma = (0,0,0)
\end{array}
\qquad (A.1)
$$

$$
\begin{array}{l}
\text{A.1.2 Case } |M_{0,4}(\frac{1}{2}, \frac{1}{2}, 0, 0)| = 1 \\
\qquad \begin{array}{c} \bullet \\ \bullet \end{array} \quad \begin{array}{c} \times \\ \times \end{array} \\
\sigma = (0, \frac{1}{2}, \frac{1}{2})
\end{array}
\qquad (A.2)
$$
A.1.8 Case $|M_{0,4}(2,0,0,0)| = 6$

<figure><img src="image_30.png" /></figure>

A.1.9 Case $|M_{0,4}(1, 1, \frac{1}{2}, \frac{1}{2})| = 3$

<figure><img src="image_31.png" /></figure>

A.1.10 Case $|M_{0,4}(1, 1, 1, 0)| = 5$

<figure><img src="image_32.png" /></figure>

A.1.11 Case $|M_{0,4}(\frac{3}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2})| = 5$

<figure><img src="image_33.png" /></figure>

A.1.12 Case $|M_{0,4}(\frac{3}{2}, \frac{1}{2}, 1, 0)| = 4$

<figure><img src="image_34.png" /></figure>
A.1.8 Case $|M_{0,4}(2,0,0,0)| = 6$

<figure><img src="image_24.png" /></figure>

A.1.9 Case $|M_{0,4}(1,1,\frac{1}{2},\frac{1}{2})| = 3$

<figure><img src="image_25.png" /></figure>

A.1.10 Case $|M_{0,4}(1,1,1,0)| = 5$

<figure><img src="image_26.png" /></figure>

A.1.11 Case $|M_{0,4}(\frac{3}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})| = 5$

<figure><img src="image_27.png" /></figure>

A.1.12 Case $|M_{0,4}(\frac{3}{2},\frac{1}{2},1,0)| = 4$

<figure><img src="image_28.png" /></figure>
A.1.13 Case $|\mathcal{M}_{0,4}(\frac{3}{2}, \frac{3}{2}, 0, 0)| = 5$

<figure><img src="image_35.png" /></figure>

A.1.14 Case $|\mathcal{M}_{0,4}(2, \frac{1}{2}, \frac{1}{2}, 0)| = 5$

<figure><img src="image_36.png" /></figure>

A.1.15 Case $|\mathcal{M}_{0,4}(2, 1, 0, 0)| = 7$

<figure><img src="image_37.png" /></figure>

A.1.16 Case $|\mathcal{M}_{0,4}(\frac{5}{2}, \frac{1}{2}, 0, 0)| = 7$

<figure><img src="image_38.png" /></figure>

A.1.17 Case $|\mathcal{M}_{0,4}(3, 0, 0, 0)| = 11$

<figure><img src="image_39.png" /></figure>
A.1.13 Case $|\mathcal{M}_{0,4}(\frac{3}{2}, \frac{3}{2}, 0, 0)| = 5$

<figure><img src="image_29.png" /></figure>

$$ \sigma = (0, \frac{3}{2}, \frac{3}{2}) \qquad \sigma = (1, \frac{3}{2}, \frac{3}{2}) \qquad \sigma = (2, \frac{1}{2}, \frac{5}{2}) $$
$$ m = 2 \qquad m = 2 \qquad m = 2 $$

(A.13)

A.1.14 Case $|\mathcal{M}_{0,4}(2, \frac{1}{2}, \frac{1}{2}, 0)| = 5$

<figure><img src="image_30.png" /></figure>

$$ \sigma = (\frac{5}{2}, 0, \frac{5}{2}) \qquad \sigma = (\frac{3}{2}, 1, \frac{3}{2}) \qquad \sigma = (\frac{1}{2}, 2, \frac{3}{2}) $$
$$ m = 2 \qquad m = 2 \qquad m = 2 $$

(A.14)

A.1.15 Case $|\mathcal{M}_{0,4}(2, 1, 0, 0)| = 7$

<figure><img src="image_31.png" /></figure>

$$ \sigma = (0, 2, 2) \qquad \sigma = (3, 0, 3) \qquad \sigma = (1, 1, 2) \qquad \sigma = (2, 1, 2) $$
$$ m = 2 \qquad m = 2 \qquad m = 2 \qquad m = 2 $$

(A.15)

A.1.16 Case $|\mathcal{M}_{0,4}(\frac{5}{2}, \frac{1}{2}, 0, 0)| = 7$

<figure><img src="image_32.png" /></figure>

$$ \sigma = (0, \frac{5}{2}, \frac{5}{2}) \qquad \sigma = (2, \frac{1}{2}, \frac{5}{2}) \qquad \sigma = (1, \frac{3}{2}, \frac{5}{2}) \qquad \sigma = (2, \frac{3}{2}, \frac{3}{2}) $$
$$ m = 2 \qquad m = 2 \qquad m = 2 \qquad m = 2 $$

(A.16)

A.1.17 Case $|\mathcal{M}_{0,4}(3, 0, 0, 0)| = 11$

<figure><img src="image_33.png" /></figure>

$$ \sigma = (2, 2, 2) \qquad \sigma = (0, 3, 3) \qquad \sigma = (1, 2, 3) $$
$$ m = 2 \qquad m = 3 \qquad m = 6 $$

(A.17)
## A.2 All connected planar maps with four or five edges

$$ \text{A.2.1 Case } |\mathcal{M}_{0,4}^c(1, 1, 1, 1)| = 3 \qquad \sigma = (1, 1, 2) \qquad \sigma = (2, 1, 1) \qquad \sigma = (1, 2, 1) \tag{A.18} $$

<figure><img src="image_40.png" /></figure>

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

$$ \text{A.2.2 Case } |\mathcal{M}_{0,4}^c(\frac{3}{2}, 1, 1, \frac{1}{2})| = 4 \qquad \sigma = (\frac{3}{2}, 1, \frac{3}{2}) \qquad \sigma = (\frac{3}{2}, 1, \frac{3}{2}) \qquad \sigma = (\frac{1}{2}, 2, \frac{3}{2}) \qquad \sigma = (\frac{3}{2}, 2, \frac{1}{2}) \tag{A.19} $$

<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.png" /></figure>

<figure><img src="image_45.png" /></figure>

<figure><img src="image_46.png" /></figure>

$$ \text{A.2.3 Case } |\mathcal{M}_{0,4}^c(\frac{3}{2}, \frac{3}{2}, \frac{1}{2}, \frac{1}{2})| = 4 \qquad \sigma = (1, 1, 2) \qquad \sigma = (2, 1, 2) \qquad \sigma = (1, 2, 1) \qquad \sigma = (2, 2, 1) \tag{A.20} $$

<figure><img src="image_47.png" /></figure>

<figure><img src="image_48.png" /></figure>

<figure><img src="image_49.png" /></figure>

<figure><img src="image_50.png" /></figure>

$$ \text{A.2.4 Case } |\mathcal{M}_{0,4}^c(2, 1, \frac{1}{2}, \frac{1}{2})| = 5 \qquad \sigma = (1, \frac{3}{2}, \frac{3}{2}) \qquad \sigma = (2, \frac{1}{2}, \frac{5}{2}) \qquad \sigma = (2, \frac{3}{2}, \frac{3}{2}) \tag{A.21} $$

<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /></figure>

<figure><img src="image_53.png" /></figure>

$$ \text{A.2.5 Case } |\mathcal{M}_{0,4}^c(\frac{5}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2})| = 6 \qquad \sigma = (2, 1, 2) \qquad \sigma = (2, 1, 2) \tag{A.22} $$

<figure><img src="image_54.png" /></figure>

<figure><img src="image_55.png" /></figure>

$$ m = 3 $$

$$ m = 3 $$
## A.2 All connected planar maps with four or five edges

$$ \text{A.2.1 Case } |\mathcal{M}_{0,4}^c(1,1,1,1)| = 3 \qquad \sigma = (1,1,2) \qquad \sigma = (2,1,1) \qquad \sigma = (1,2,1) \tag{A.18} $$

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /></figure>

<figure><img src="image_36.png" /></figure>

$$ \text{A.2.2 Case } |\mathcal{M}_{0,4}^c(\frac{3}{2},1,1,\frac{1}{2})| = 4 \qquad \sigma = (\frac{3}{2}, 1, \frac{3}{2}) \qquad \sigma = (\frac{3}{2}, 1, \frac{3}{2}) \qquad \sigma = (\frac{1}{2}, 2, \frac{3}{2}) \qquad \sigma = (\frac{3}{2}, 2, \frac{1}{2}) \tag{A.19} $$

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

$$ \text{A.2.3 Case } |\mathcal{M}_{0,4}^c(\frac{3}{2}, \frac{3}{2}, \frac{1}{2}, \frac{1}{2})| = 4 \qquad \sigma = (1, 1, 2) \qquad \sigma = (2, 1, 2) \qquad \sigma = (1, 2, 1) \qquad \sigma = (2, 2, 1) \tag{A.20} $$

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.png" /></figure>

$$ \text{A.2.4 Case } |\mathcal{M}_{0,4}^c(2,1,\frac{1}{2},\frac{1}{2})| = 5 \qquad \sigma = (1, \frac{3}{2}, \frac{3}{2}) \qquad \sigma = (2, \frac{1}{2}, \frac{5}{2}) \qquad \sigma = (2, \frac{3}{2}, \frac{3}{2}) \tag{A.21} $$

<figure><img src="image_45.png" /></figure>

<figure><img src="image_46.png" /></figure>

<figure><img src="image_47.png" /></figure>

$$ \text{A.2.5 Case } |\mathcal{M}_{0,4}^c(\frac{5}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2})| = 6 \qquad \sigma = (2, 1, 2) \qquad \sigma = (2, 1, 2) \tag{A.22} $$

<figure><img src="image_48.png" /></figure>

<figure><img src="image_49.png" /></figure>

$$ \sigma = (2, 1, 2) \qquad \sigma = (2, 1, 2) $$

$$ m = 3 $$
A.2.6 Case $|M_{0,4}^c(\frac{3}{2}, \frac{3}{2}, 1, 1)| = 6$

<figure><img src="image_56.png" /></figure>

$$ \sigma = \left(1, \frac{3}{2}, \frac{5}{2}\right) \\ m = 2 $$

<figure><img src="image_57.png" /></figure>

$$ \sigma = \left(2, \frac{3}{2}, \frac{3}{2}\right) \\ m = 2 $$

<figure><img src="image_58.png" /></figure>

$$ \sigma = \left(2, \frac{1}{2}, \frac{5}{2}\right) \\ m = 2 $$

(A.23)

A.2.7 Case $|M_{0,4}^c(\frac{3}{2}, \frac{3}{2}, \frac{3}{2}, \frac{1}{2})| = 6$

<figure><img src="image_59.png" /></figure>

$$ \sigma = (2, 2, 1) \\ m = 3 $$

<figure><img src="image_60.png" /></figure>

$$ \sigma = (2, 2, 1) \\ m = 3 $$

(A.24)

A.2.8 Case $|M_{0,4}^c(2, 1, 1, 1)| = 6$

<figure><img src="image_61.png" /></figure>

$$ \sigma = (1, 2, 2) \\ m = 3 $$

<figure><img src="image_62.png" /></figure>

$$ \sigma = (1, 2, 2) \\ m = 3 $$

(A.25)

A.2.9 Case $|M_{0,4}^c(2, \frac{3}{2}, \frac{1}{2}, 1)| = 7$

<figure><img src="image_63.png" /></figure>

$$ \sigma = (\frac{1}{2}, 2, \frac{5}{2}) $$

<figure><img src="image_64.png" /></figure>

$$ \sigma = (\frac{3}{2}, 1, \frac{5}{2}) $$

<figure><img src="image_65.png" /></figure>

$$ \sigma = (\frac{5}{2}, 1, \frac{5}{2}) $$

<figure><img src="image_66.png" /></figure>

$$ \sigma = (\frac{3}{2}, 2, \frac{3}{2}) $$

<figure><img src="image_67.png" /></figure>

$$ \sigma = (\frac{3}{2}, 2, \frac{3}{2}) $$

<figure><img src="image_68.png" /></figure>

$$ \sigma = (\frac{5}{2}, 2, \frac{3}{2}) $$

<figure><img src="image_69.png" /></figure>

$$ \sigma = (\frac{5}{2}, 3, \frac{1}{2}) $$

(A.26)
A.2.10 Case $|\mathcal{M}_{0,4}^c(2, 2, \frac{1}{2}, \frac{1}{2})| = 8$

<figure><img src="image_70.png" /></figure>

A.2.11 Case $|\mathcal{M}_{0,4}^c(\frac{5}{2}, 1, 1, \frac{1}{2})| = 8$

<figure><img src="image_71.png" /></figure>

A.2.12 Case $|\mathcal{M}_{0,4}^c(\frac{5}{2}, \frac{3}{2}, \frac{1}{2}, \frac{1}{2})| = 8$

<figure><img src="image_72.png" /></figure>

A.2.13 Case $|\mathcal{M}_{0,4}^c(3, 1, \frac{1}{2}, \frac{1}{2})| = 10$

<figure><img src="image_73.png" /></figure>

A.2.14 Case $|\mathcal{M}_{0,4}^c(\frac{7}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2})| = 12$

<figure><img src="image_74.png" /></figure>
A.2.10 Case $|\mathcal{M}_{0,4}^c(2, 2, \frac{1}{2}, \frac{1}{2})| = 8$

<figure><img src="image_50.png" /></figure>

A.2.11 Case $|\mathcal{M}_{0,4}^c(\frac{5}{2}, 1, 1, \frac{1}{2})| = 8$

<figure><img src="image_51.png" /></figure>

A.2.12 Case $|\mathcal{M}_{0,4}^c(\frac{5}{2}, \frac{3}{2}, \frac{1}{2}, \frac{1}{2})| = 8$

<figure><img src="image_52.png" /></figure>

A.2.13 Case $|\mathcal{M}_{0,4}^c(3, 1, \frac{1}{2}, \frac{1}{2})| = 10$

<figure><img src="image_53.png" /></figure>

A.2.14 Case $|\mathcal{M}_{0,4}^c(\frac{7}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2})| = 12$

<figure><img src="image_54.png" /></figure>
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