Linking numbers of modular knots

TL;DR

This work links the topology of modular knots to the arithmetic of PSL$(2,\mathbb{Z})$ by showing that modular-linking numbers with the trefoil are captured by the Rademacher invariant and by constructing a rich family of linking functionals on the PSL$(2,\mathbb{Z})$ character variety. It develops a deformation framework $\rho_q$ that interpolates between hyperbolic-geometric data and tree-based combinatorics, defines L$q$ and C$q$ as boundary-valued invariants, and shows their limits recover classical linking data while revealing pole/zero concentration on the unit circle. The paper also provides concrete combinatorial and algebraic formulas for linking numbers via double cosets, introduces a basis of homogeneous quasi-morphisms $\mathrm{C}_A$ tied to primitive conjugacy classes, and proves the nondegeneracy of the linking form, establishing a robust bridge between dynamics, topology, and arithmetic. These results open routes to relate modular-knot linking to Alexander-type invariants, Poincaré series, and bounded cohomology, with broad implications for arithmetic and geometric deformations of Fuchsian groups.

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.

Figures (10)

• Figure : Angle $\theta\in \,]0,\pi[$. Ortho-geodesics: co-oriented ($\lambda>0$) and disco-oriented ($\lambda<0$).
• Figure : The ideal triangulation of $\mathbb{H}\mathbb{P}$ together with its dual trivalent tree $\mathcal{T}$ yield the modular tessellation with fundamental domain $(0,j,\infty)$.
• Figure : The geometric axes in $\mathbb{P}\mathbb{H}$ and their projections in $\mathbb{M}$ of $RL$, $RLL$, $RLLL$.
• Figure : Configurations of axes: $\mathop{\mathrm{cross}}\nolimits$ and $\mathop{\mathrm{cosign}}\nolimits$. Note that $\mathop{\mathrm{cross}}\nolimits\ne 0 \implies \mathop{\mathrm{cosign}}\nolimits = \pm 1$.
• Figure : The convex core of $\mathbb{M}_q$ lifts in $\mathbb{H}\mathbb{P}$ to an $\epsilon$-neighbourhood of $\mathcal{T}_q$ with $\epsilon= \Theta\left(1/q^2\right)$.

Theorems & Definitions (41)

• Theorem 1: Linking and intersection from boundary evaluations
• Proposition 2: Alexander, Fricke and Rademacher
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 1.1: CLS_Conj-PSL2K_2022
• Lemma 1.2: Real geometry of the cross-ratio
• proof
• Proposition 3.1
• proof