Quadratic irrationals and linking numbers of modular knots

Dubi Kelmer

doi:10.3934/jmd.2012.6.539

Article Contents

2012, Volume 6, Issue 4: 539-561. Doi: 10.3934/jmd.2012.6.539

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Quadratic irrationals and linking numbers of modular knots

Dubi Kelmer^1,

1.
Boston College, Department of Mathematics, Chestnut Hill, MA 02467

Received: May 31, 2012

Published: December 2012

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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.

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Mathematics Subject Classification: Primary: 58F17; Secondary: 11F72, 30B70, 58F20.

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