Let me define a **degree $n$ colored arrangement of circles** on $S^2$ to be a collection $\mathcal{C}$ of $n$ disjoint, smoothly embedded circles $C_1,\dotsc, C_n\subset S^2$ together with a continuous map (temperature)

$$ T: S^2 \setminus (C_1\cup\cdots \cup C_n)\to \lbrace 1,-1\rbrace $$

such that for any circle $C_k$, the temperatures on opposite sides of $C_k$ are different. In other words, if $\gamma:(-\varepsilon,\varepsilon)\to S^2$, $t\mapsto \gamma(t)$, is a short smooth path that intersects exactly one circle $C_k$, and it does so transversally at $t=0$, then

$$\lim_{t\searrow 0} \; T\bigl(\; \gamma(t)\;\bigr)= - \lim_{t\nearrow 0}\; T\bigl(\;\gamma(t)\;\bigr). $$

Think of the components of $T^{-1}(-1)$ as *icy regions* and of the components of $T^{-1}(1)$ as *liquid regions*. We will refer to $T$ as a *coloring* of the *arrangement of circles* $\mathcal{C}$.

Two arrangements of circles $\mathcal{C}$ and $\mathcal{C}'$ are called **equivalent** if there exists an orientation preserving diffeomorphism of $S^2$ that maps one arrangement to the other. We denote by $\mathcal{A}_n$ the set of equivalence classes of arrangements on $n$ disjoint circles on $S^2$.

Two degree $n$ colored arrangements $(\mathcal{C}, T)$ and $(\mathcal{C}', T')$ are called **equivalent** if there exists an orientation preserving diffeomorphism $\Psi: S^2\to S^2$ mapping the circles in $\mathcal{C}$ to the circles in $\mathcal{C}'$ and such that $T'\circ \Psi= T$.

Two colorings of a given arrangement $\mathcal{C}$ of $n$ disjoint circles on $S^2$ are called **equivalent** if the colored arrangements $(\mathcal{C}, T)$ and $(\mathcal{C}, T')$ are equivalent. We get in this fashion a map

$$ K_n : \mathcal{A}_n\to\mathbb{Z}$$

that associates to an arrangement $\mathcal{C}$ the nummber of inequivalent colorings of $\mathcal{C}$. Here are some questions I find interesting.

$\mathbf{Q_0}$ Find the cardinality of $\mathcal{A}_n$ ans its large $n$ asymptotics.

$\mathbf{Q_1}$ Find the cardinality of $\mathcal{S}_n$ and its large $n$ asymptotics.

$\mathbf{Q_1^*}$ Investigate the map $K_n$.

$\mathbf{Q_2}$ We declare an colored arrangement $(\mathcal{C}, T)$ to be **selfdual** if it is equivalent to $(\mathcal{C}, -T)$. Find the number of selfdual colored arrangements of degree $n$ and its large $n$ asymptotics.

$\mathbf{Q_3}$ Equip $\mathcal{S}_n$ with the uniform probability density and denote by $X_n:\mathcal{S}_n\to \mathbb{R}$ the random variable o that associates to a colored arrangement $(\mathcal{C}, T)$ the Euler characteristic of the liquid region $T^{-1}(1)$. Denote by $\mu_n$ the probability distribution of $X_n$. Investigate the large $n$ behavior of $\mu_n$.

$\mathbf{Q_4}$ Can you think of a good notion of a random arrangement of $n$ disjoint circles on $S^2$?