Canonical embeddings of pairs of arcs

TL;DR

This paper proves the existence and uniqueness of a canonical embedding of pairs of arcs in the Riemann sphere, where each arc is a hyperbolic geodesic in the complement of the other, given four points and an isotopy class.

Contribution

It establishes a unique canonical configuration for pairs of arcs connecting four points on the Riemann sphere with hyperbolic geodesic properties.

Findings

Existence of a unique configuration of arcs as hyperbolic geodesics.

Canonical embedding characterized by hyperbolic geometry.

Applicable to isotopy classes of arc pairs in the Riemann sphere.

Abstract

We show that for given four points in the Riemann sphere and a given isotopy class of two disjoint arcs connecting these points in two pairs, there exists a unique configuration with the property that each arc is a hyperbolic geodesic segment in the complement of the other arc.