Knots, Perturbative Series and Quantum Modularity

We introduce an invariant of a hyperbolic knot which is a map _ (h) from Q/Z to matrices with entries in Q[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 (₀, ₁) entry, where ₀ is the trivial and ₁ the geometric representation, is the power series expansion of the Kashaev invariant of the knot around the root of unity e^2 i 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 are fundamental solutions of a linear q-difference equation; (d) the matrix defines an SL₂ (Z) -cocycle W_ 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 (h) to actual functions. The two invariants and W_ 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.