From 3-dimensional skein theory to functions near Q

TL;DR

This paper constructs a map from skein modules of 3-manifolds to the Habiro ring, linking quantum topology with number theory and providing a framework for understanding quantum modularity.

Contribution

It introduces a new map connecting skein modules to the Habiro ring, enabling the study of quantum modularity in a number-theoretic context.

Findings

The image of the map is a finitely generated module.

The construction relates skein theory to quantum modular forms.

It offers a new approach to the Quantum Modularity Conjecture.

Abstract

Motivated by the Quantum Modularity Conjecture and its arithmetic aspects related to the Habiro ring of a number field, we define a map from the Kauffman bracket skein module of an integer homology 3-sphere to the Habiro ring, and use Witten's conjecture (now a theorem) to show that the image is an effectively computable module of finite rank that can be used to phrase the quantum modularity conjecture.