Linking numbers of modular knots

Simon, Christopher-Lloyd

doi:10.2140/gt.2025.29.3241

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Linking numbers of modular knots

Christopher-Lloyd Simon

Geometry & Topology 29 (2025) 3241–3270

DOI: 10.2140/gt.2025.29.3241

Abstract

The modular group ${PSL}_{2} (ℤ)$ acts on the hyperbolic plane $ℍ ℙ$ 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 sphere $𝕊^{3}$ . 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 are well understood. Indeed, É Ghys showed in 2006 that they are given by the Rademacher invariant of the corresponding conjugacy classes. The Rademacher function $Rad : {PSL}_{2} (ℤ) \to ℤ$ is a pseudocharacter, which he recognised with J Barge in 1992 as half the primitive of the bounded Euler class.

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 pseudocharacter on the modular group, and how to extract out of these a Schauder basis for the Banach space of pseudocharacters.

Keywords

modular group, modular geodesics, intersection numbers, Lorenz template, linking numbers, bounded cohomology, $q$-analogues, knot polynomials

Mathematical Subject Classification

Primary: 11F06, 57K10, 57K14, 57K20, 57K31

Secondary: 20H10

References

Publication

Received: 19 December 2022

Revised: 26 January 2024

Accepted: 4 March 2024

Published: 22 September 2025

Proposed: Cameron Gordon

Seconded: Anna Wienhard, Mladen Bestvina

Authors

Christopher-Lloyd Simon
Department of Mathematics Eberly College of Science The Pennsylvania State University State College, PA United States

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Volume 29, issue 6 (2025)