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 ${\text{SL}}_{2}Z$ . In this article, we replace ${\text{SL}}_{2}Z={\Gamma }_{2,3}$ 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.