Quadratic irrationals and linking numbers of modular knots

TL;DR

This paper establishes that closed geodesics on the modular surface with a fixed linking number become equidistributed on average, using thermodynamic formalism to connect geometric properties with quadratic irrationals.

Contribution

It introduces a novel connection between linking numbers of modular knots and quadratic irrationals, proving their equidistribution with respect to length and linking number.

Findings

Closed geodesics with prescribed linking numbers are equidistributed on average.

The set of quadratic irrationals corresponding to these geodesics also exhibits equidistribution.

Thermodynamic formalism is used to establish these distribution results.

Abstract

A closed geodesic on the modular surface gives rise to a knot on the 3-sphere with a trefoil knot removed, and one can compute the linking number of such a knot with the trefoil knot. We show that, when ordered by their length, the set of closed geodesics having a prescribed linking number become equidistributed on average with respect to the Liouville measure. We show this by using the thermodynamic formalism to prove an equidistribution result for a corresponding set of quadratic irrationals on the unit interval.