Knots, Perturbative Series and Quantum Modularity

Stavros Garoufalidis, Don Zagier

We introduce an invariant of a hyperbolic knot which is a map $\hat{I}\pm \mapsto \hat{\mathbf{I}}{¦}_{\hat{I}}\pm (h)$ from $\mathbb{Q}/\mathbb{Z}$ to matrices with entries in $\overline{\mathbb{Q}}[[h]]$…