A lower bound for the volumes of complements of periodic geodesics

TL;DR

This paper establishes a lower bound on the hyperbolic volume of knot complements associated with filling closed geodesics on surfaces, relating it to the topology of the geodesic and surface decomposition.

Contribution

It introduces a novel lower bound for the volume of these knot complements based on the homotopy classes of arcs in a pants decomposition.

Findings

Provides explicit lower bounds for volumes of knot complements

Relates geometric volume to topological features of geodesics

Advances understanding of hyperbolic structures in surface topology

Abstract

Every closed geodesic $\gamma$ on a surface has a canonically associated knot $\widehat\gamma$ in the projective unit tangent bundle. We study, for $\gamma$ filling, the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We provide a lower bound for the volume relative to the number of homotopy classes of $\gamma$-arcs in each pair of pants of a pants decomposition of the surface.