Knots, Perturbative Series and Quantum Modularity

We introduce an invariant of a hyperbolic knot which is a map α↦ Φ_α(h) from Q/Z to matrices with entries in Q̄[[h]] and with rows and columns indexed by the boundary parabolic SL₂(C) representations of the fundamental group of the knot. These matrix invariants have a rich structure: (a) their (σ₀,σ₁) entry, where σ₀ is the trivial and σ₁ the geometric representation, is the power series expansion of the Kashaev invariant of the knot around the root of unity e^2π i α 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; (c) the columns of Φ are fundamental solutions of a linear q-difference equation; (d) the matrix defines an SL₂(Z)-cocycle W_γ in matrix-valued functions on Q that conjecturally extends to a smooth function on R and even to holomorphic functions on suitable complex cut planes, lifting the factorially divergent series Φ(h) to actual functions. The two invariants Φ and W_γ are related by a refined quantum modularity conjecture which we illustrate in detail for the three simplest hyperbolic knots, the 4₁, 5₂ 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.