Canonical embeddings of pairs of arcs (Q2129491)

The authors consider chord diagrams in the closure \(\overline{\mathbb D}\) of the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\) which have prescribed points \(a_1,\dots,a_{2d-2}\), \(d\geq2\), on the unit circle and \(d-1\) disjoint crosscuts \(e_1,\dots,e_{d-1}\) in \(\mathbb D\) connecting pairs of these points. Such a chord diagram is canonical if every crosscut \(e_k\) is a hyperbolic geodesic in the unique component of \(\mathbb D\setminus\bigcup_{j\neq k}e_j\) that contains the interior points of \(e_k\). In the paper, there is an example with four prescribed points and infinitely many canonical configurations. The configurations consist of four distinct points \(a_0,a_1,a_2,a_3\) in the Riemann sphere \(\hat{\mathbb C}=\mathbb C\cup\{\infty\}\) and two disjoint arcs \(\gamma_0\) and \(\gamma_1\), where \(\gamma_0\) has the endpoints \(a_0\) and \(a_1\), and \(\gamma_1\) has the endpoints \(a_2\) and \(a_3\). Two such configurations are equivalent if the points are the same, and the arcs of the first configuration can be deformed into the arcs of the second configuration by an isotopy of the sphere that keeps the endpoints of the arcs fixed. A configuration is canonical if for each \(k\in\{0,1\}\) the arc \(\gamma_k\) is a hyperbolic geodesic segment in the simply connected region \(\hat{\mathbb C}\setminus\gamma_{1-k}\). The authors prove the following theorem. Theorem: For every equivalence class of configurations, there exists a unique canonical configuration.