Skeins, $q$-series, and modularity

TL;DR

This paper investigates BPS q-series linked to 3-manifolds with embedded links, establishing their dependence on skein classes and proposing a connection to quantum modular forms and Langlands duality.

Contribution

It proves that BPS q-series depend only on skein classes, defining a homomorphism from skein modules to q-series, and conjectures their quantum modularity.

Findings

BPS q-series depend solely on skein class of the link

A homomorphism from skein module to q-series is established

Conjecture: the image is holomorphically quantum modular

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.