# $\hat{Z}$ invariants at rational $\tau$

## TL;DR

This paper explores the relationship between $ ext{\hat{Z}}$ invariants and Reshetikhin-Turaev invariants at rational values of $ au$, proposing a conjecture supported by tests on different 3-manifolds.

## Contribution

It introduces a conjecture linking $ ext{\hat{Z}}$ invariants with Reshetikhin-Turaev invariants at rational $ au$, extending their modularity properties.

## Key findings

- Connected Reshetikhin-Turaev and $ ext{\hat{Z}}$ invariants at rational $ au$
- Proposed a conjecture supported by tests on various 3-manifolds
- Highlighted modularity properties of $ ext{\hat{Z}}$ invariants

## Abstract

$\hat{Z}$ invariants of 3-manifolds were introduced as series in $q=e^{2\pi i\tau}$ in order to categorify Witten-Reshetikhin-Turaev invariants corresponding to $\tau=1/k$. However modularity properties suggest that all roots of unity are on the same footing. The main result of this paper is the expression connecting Reshetikhin-Turaev invariants with $\hat{Z}$ invariants for $\tau\in\mathbb{Q}$. We present the reasoning leading to this conjecture and test it on various 3-manifolds.