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

TL;DR

This paper generalizes Ghys's work by defining Rademacher symbols for triangle groups and relating them to linking numbers of knots, extending the connection between automorphic forms and knot theory beyond the classical modular case.

Contribution

It introduces Rademacher symbols for triangle groups using harmonic Maass forms and generalizes Ghys's theorem to a broader class of knots in 3-manifolds.

Findings

Defined Rademacher symbols for triangle groups ,p,q

Connected Rademacher symbols to linking numbers of knots

Extended Ghys's theorem to torus knots in lens spaces

Abstract

\'{E}.\~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 ${\rm SL}_2\mathbb{Z}$. In this paper, we replace ${\rm SL}_2\mathbb{Z}=\Gamma_{2,3}$ by the triangle group $\Gamma_{p,q}$ for any coprime pair $(p,q)$ of integers with $2\leq p<q$. 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.