Consider configurations consisting of 4 distinct circles on the sphere.
Two configurations are equivalent if they can be mapped onto each other by a homeomorphism of the sphere. How to enumerate/classify such configurations?

Equivalent problem: classify the arrangements of 4 hyperbolic planes in the hyperbolic space, up to homeomorphisms of the space.

Before voting to close this question as trivial, you may look at the <a href="http://www.math.purdue.edu/~eremenko/dvi/4circles-generic.pdf">classification of generic configurations</a> which we obtained by brute force:

[![configurations without disjoint pairs of circles][1]][1]

[![configurations with at least one disjoint pair of circles][2]][2]

[Each region bounded by more than 3 sides is labeled by the number of its boundary sides.]

Questions: Is this new? Is there a scientific method to obtain this? Is there any structure on these 35 configurations?

There is a large research area about hyperplane arrangements in a Euclidean space. How about hyperbolic space? There is also a large body of research on
hyperbolic tetrahedra. But it is always assumed that the tetrahedron is compact (or has only vertices at infinity).

We encountered this question in <a href="http://www.math.purdue.edu/~eremenko/dvi/pent18.pdf">our studies of the Heun and Painlevé VI equations</a> with real coefficients. (See Appendix II).
Projective monodromy groups associated to these equations are generated by 4 reflections in circles.


  [1]: https://i.sstatic.net/Qul0y.png
  [2]: https://i.sstatic.net/cnhTI.png