Equivariant Topology of Configuration Spaces of the Circle

This work was firstly inspired by Mostovoy's paper, where he showed Bott's construction proving that a space of non-empty finite subsets of circle of cardinality at most 3, denoted $exp_3(S^1)$ (and called potency space) is homeomorphic to $S^3$. And Schepin's theorem claims that the subspace of one-element subsets $exp_1(S^1)$ is a trefoil knot $K$ on that sphere. All such inclusions correspond to a moduli space of elliptic curves, which is, on the other hand obtained as the space of all lattices ($PSL(2, \mathbb{R})$) quotient by a discrete subgroup $PSL(2, \mathbb{Z})$. The question I was interested in is if there're some other examples, where this space arises. On the other hand, my main goal was to try to make a conjecture if it's possible to obtain some generalizations of connections between potency spaces of circle with more marked points and some moduli spaces.