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 in coincide with the values of a highly ubiquitous function called the Rademacher symbol for . In this article, we replace by the triangle group for any coprime pair ( p , q ) of integers with . We invoke the theory of harmonic Maass forms for to introduce the notion of the Rademacher symbol , and provide several characterizations. Among other things, we generalize Ghys’s theorem for modular knots around any “missing” torus knot in and in a lens space.