Skeins, $q$-series, and modularity

Abstract: We study BPS $q$-series associated to 3-manifolds decorated by a line defect along an embedded link. We prove that these $q$-series depend only on the class of the link in the skein module, thereby defining a homomorphism from the skein module to the space of $q$-series. The image of this homomorphism is conjectured to be holomorphically quantum modular, which suggests a new approach to Langlands duality for skein modules through $q$-series.