Modular knots, automorphic forms, and the Rademacher symbols for triangle groups

Abstract

É.Ghys proved that the linking numbers of modular knots and the "missing" trefoil $K_{2,3}$ in $S^3$ coincide with the values of a highly ubiquitous function called the Rademacher symbol for ${\mathrm{SL}}_2\mathbb{Z}$. In this paper, we replace ${\mathrm{SL}}_2\mathbb{Z}={\Gamma }_{2,3}$ by the triangle group ${\Gamma }_{p,q}$ for any coprime pair $(p,q)$ of integers with $2\le p<q$. We invoke the theory of harmonic Maass forms for ${\Gamma }_{p,q}$ to introduce the notion of the Rademacher symbol ${\psi }_{p,q}$, and provide several characterizations. Among other things, we generalize Ghys's theorem for modular knots around any "missing" torus knot $K_{p,q}$ in $S^3$ and in a lens space.

34 pages, 1 figure; v2: Section 5 is refined; The Sarnak-Mozzochi formula for ${\Gamma }_{p,q}$ is added

References in corpus (2)

• Computing intersections of closed geodesics on the modular curve
• The Manin-Drinfeld theorem and the rationality of Rademacher symbols

Cited by in corpus (2)

• A Hyperbolic Analogue of the Rademacher Symbol
• Windings of prime geodesics