Knots, Perturbative Series and Quantum Modularity

We introduce an invariant of a hyperbolic knot which is a map $\alpha\mapsto \boldsymbol{\Phi}_\alpha(h)$from$$\mathbb{Q}$/$\mathbb{Z}$$ to matrices with entries in $\bar{\mathbb{Q}}[[h]]$ and with rows and columns indexed by the boundary parabolic ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ representations of the fundamental group of the knot. These matrix invariants have a rich structure: (a) their $({\sigma }_0,{\sigma }_1)$ entry, where ${\sigma }_0$ is the trivial and ${\sigma }_1$ the geometric representation, is the power series expansion of the Kashaev invariant of the knot around the root of unity ${\rm e}^{2\pi{\rm i} \alpha}$ as an element of the Habiro ring, and the remaining entries belong to generalized Habiro rings of number fields; (b) the first column is given by the perturbative power series of Dimofte-Garoufalidis; © the columns of $\mathbf{\Phi }$ are fundamental solutions of a linear $q$-difference equation; (d) the matrix defines an ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$-cocycle $W_{\gamma }$ in matrix-valued functions on $\mathbb{Q}$ that conjecturally extends to a smooth function on $\mathbb{R}$ and even to holomorphic functions on suitable complex cut planes, lifting the factorially divergent series $\mathbf{\Phi }(h)$ to actual functions. The two invariants $\mathbf{\Phi }$ and $W_{\gamma }$ are related by a refined quantum modularity conjecture which we illustrate in detail for the three simplest hyperbolic knots, the $4_1$, $5_2$ and $(-2,3,7)$ pretzel knots. This paper has two sequels, one giving a different realization of our invariant as a matrix of convergent $q$-series with integer coefficients and the other studying its Habiro-like arithmetic properties in more depth.