Linking numbers of modular knots

TL;DR

This paper explores the linking numbers of modular knots in the hyperbolic 3-manifold related to the modular group, deriving formulas and establishing their connection to quasi-morphisms and character varieties.

Contribution

It introduces new formulas for linking numbers of modular knots and shows how these relate to quasi-morphisms on the modular group, expanding understanding of their algebraic and topological properties.

Findings

Linking numbers are expressed via arithmetical, combinatorial, and topological formulas.

The difference of linking numbers with a knot and its inverse forms a homogeneous quasi-morphism.

The linking pairing is proven to be non-degenerate, enabling the extraction of a free basis.

Abstract

The modular group PSL(2;Z) acts on the hyperbolic plane HP with quotient the modular surface M, whose unit tangent bundle U is a 3-manifold homeomorphic to the complement of the trefoil knot in the 3-sphere. The hyperbolic conjugacy classes of PSL(2;Z) correspond to the closed oriented geodesics in M. Those lift to the periodic orbits for the geodesic flow in U, which define the modular knots. The linking numbers between modular knots and the trefoil is well understood. Indeed, Etienne Ghys showed in 2006 that they are given by the Rademacher invariant of the corresponding conjugacy classes. The Rademacher function is a homogeneous quasi-morphism of PSL(2;Z) which he had recognised with Jean Barge in 1992 as half the primitive of the bounded euler class. This shed light on the 1987 work of Michael Atiyah concerning the logarithm of the Dedekind eta function which identified it with no less than that six other important functions appearing in diverse areas of mathematics. We are concerned with the linking numbers between modular knots and derive several formulae with arithmetical, combinatorial, topological and group theoretical flavours. In particular we associate to a pair of modular knots a function defined on the character variety of PSL(2;Z), whose limit at the boundary point recovers their linking number. Moreover, we show that the linking number with a modular knot minus that with its inverse yields a homogeneous quasi-morphism on the modular group, and how to extract a free basis out of these. For this we prove that the linking pairing is non degenerate.