Round twin groups on few strands

We study the space $Q_n$ of all configurations of $n$ ordered points on the circle such that no three points coincide, and in which one of the points (say, the last one) is fixed. We compute its fundamental group for $n<6$ and describe its homology for $n=6,7$. For arbitrary $n$, we compute its first homology and its Euler characteristic. We use three geometric approaches. On one hand, $Q_n$ is naturally defined as the complement to an arrangement of codimension-2 subtori in a real torus. On the other hand, $Q_n$ is homotopy equivalent to an explicit nonpositively curved cubical complex. Finally, $Q_n$ can also be assembled from no-3-equal manifolds of the real line. We also observe that, up to homotopy, $Q_n$ may be identified with a subspace of the oriented double cover of the moduli space $\overline{\mathcal{M}}_{0,n}(\mathbb{R})$ of stable real rational curves with $n$ marked points. This gives an embedding of $\pi_1 Q_n$ into the pure cactus group. As a corollary, we see that $\pi_1 Q_n$ is residually nilpotent.