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

Understand

$\hat{Z}$ 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.