A lower bound for the volumes of complements of periodic geodesics

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.