Abstract
Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 ( Z ) ∖ PSL 2 ( R ) . A finite collection of such orbits is a collection of disjoint closed curves in a 3-manifold, in other words a link. The complement of those links is always a hyperbolic 3-manifold, and hence has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics.
Highlights
We study numerically the topological type of knots and links arising from closed geodesics on the modular surface
√set of geodesics corresponding to the ideal classes in the ring of integers of the quadratic field Q( d)
In order to investigate the topology of the closed geodesics, we need a way to explicitly construct a representative of the isotopy class of the geodesic on the unit tangent bundle from a representation of the conjugacy class
Summary
A knot in X is a simple closed smooth curve, considered up to deformation; a link in X is a collection of such pairwise disjoint curves, again considered up to deformation. We study numerically the topological type of knots and links arising from closed geodesics on the modular surface. These geodesics have number-theoretic meaning, and we show evidence of an interesting behaviour for the links that arise when the geodesics are grouped in a number-theoretic relevant way. In the rest of this introduction, we review these general ideas, discuss the motivation for our work, and outline the rest of the paper
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