La courbe en huit sur les sph\`eres \`a pointes et le noeud de huit

TL;DR

This paper demonstrates that in a specific hyperbolic orbisphere setting, the lift of the shortest periodic geodesic is topologically equivalent to the complement of the figure-eight knot, using linking number computations.

Contribution

It establishes a novel connection between geodesic lifts in hyperbolic orbispheres and knot complements, specifically linking shortest geodesics to the figure-eight knot complement.

Findings

Lift of shortest geodesic homeomorphic to figure-eight knot complement

Linking numbers used to prove topological equivalence

Applicable to hyperbolic orbispheres with specific cone points

Abstract

We show that, in the unit tangent bundle of a hyperbolic orbisphere with cone points of order 3, 3, 4, the lift of the shortest periodic geodesic is homeomorphic to the complement of the figure-eight knot in the 3-sphere. The proof eventually relies on the computation of some linking numbers.