Periods of continued fractions and volumes of modular knots complements

Abstract

Every oriented closed geodesic on the modular surface has a canonically associated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow. We study the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We show that there exist sequences of closed geodesics for which this volume is bounded linearly in terms of the period of the geodesic’s continued fraction expansion. Consequently, we give a volume’s upper bound for some sequences of Lorenz knots complements, linearly in terms of the corresponding braid index. Also, for any punctured hyperbolic surface we give volume’s bounds for the canonical lift complement relative to some sequences of sets of closed geodesics in terms of the geodesics length.