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, and we verify this conjecture in an explicit example.

arXiv comment45 pages; accepted version