# Stratification of space of labelled circles in the plane

Consider the space of $n$ round circles in the plane to be the open subset of $\mathbb R^{3n}$:

$$C_n = \{ (v_1, v_2, \cdots, v_n, r_1, r_2, \cdots, r_n ) : v_i \in \mathbb R^2, r_i \in (0, \infty) \ \ \forall i \}$$

The $i$-th circle corresponding to a point in the above space would be the solutions to the equation $|x-v_i| = r_i$. With this convention, you can think of $C_n$ as the space of $n$ labelled circles in the plane, where the circles are allowed to intersect or even coincide.

There is a natural stratification of this space induced by the co-dimension one subvarieties consisting of subspaces where two circles are tangent, or where three (or more) circles intersect in a non-empty set. This is a fairly natural stratification since the complement of this stratification in $C_n$ is what you might call the "isotopy classes" of "regular" circles, as the combinatorics of their intersections do not change in the path-components and it is a dense open subspace of $C_n$.

Is there much known about the basic combinatorics of this stratification? For example, labelling the path-connected co-dimension $k$ strata for $k=0,1,2, \cdots$ and enumerating them? It looks like a souped-up version of the partition problem from a fairly naive perspective.

It seems like a natural problem and there is much closely related to it (spaces of arrangements in homotopy theory). But I've never come across quite this problem before.

Any insights would be appreciated -- especially if there's papers out there on this topic that I'm unaware of.

• The $v_i$ are not quite the configuration space $\text{Conf}_n(\mathbf{R}^2)$ because you are allowing circles to coincide, but also not quite the Ran space $\text{Ran}^{\leqslant n}(\mathbf{R}^2)$, because you are keeping track of how many points coincide (and their order). Ayala-Francis-Tanaka (Example 3.5.17 and onwards) write about how $X^I$, for $I$ finite, sits in an already stratified Ran space, which may give you some coarser information. – Jānis Lazovskis Feb 11 '18 at 15:35
• @JānisLazovskis: It sounds like you are referring to the codimension zero strata? There are also tangencies and triple point avoidance. – Ryan Budney Feb 14 '18 at 19:00
• Yes, as that is the closest thing I could think of of related to your question. – Jānis Lazovskis Feb 16 '18 at 17:26