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

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 SL 2 Z . In this article, we replace SL 2 Z = Γ 2 , 3 by the triangle group Γ p , q for any coprime pair (p, q) of integers with 2 ≤ p < q . We invoke the theory of harmonic Maass forms for Γ p , q to introduce the notion of the Rademacher symbol ψ 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.