The mixed mock modularity of certain duals of generalized quantum modular forms of Hikami and Lovejoy

Abstract

We resolve a question of Hikami and Lovejoy that sits at the intersection of quantum modular forms, mock modular forms, and knot theory. We express a family of Hecke–Appell-type sums of Hikami and Lovejoy in terms of mixed mock modular forms; in particular, we express the sums in terms of Appell functions and theta functions. They obtained their family of Hecke–Appell-type sums by considering certain duals of generalized quantum modular forms, where the quantum modular forms are motivated by the colored Jones polynomial for special torus knots.