  We discuss a matrix of periodic holomorphic functions in the upper and lower
half-plane which can be obtained from a factorization of an Andersen-Kashaev
state integral of a knot complement with remarkable analytic and asymptotic
properties that defines a ${\rm PSL}_2({\mathbb Z})$-cocycle on the space of
matrix-valued piecewise analytic functions on the real numbers. We identify the
corresponding cocycle with the one coming from the Kashaev invariant of a knot
(and its matrix-valued extension) via the refined quantum modularity conjecture
of [arXiv:2111.06645] and also relate the matrix-valued invariant with the
3D-index of Dimofte-Gaiotto-Gukov. The cocycle also has an analytic
extendability property that leads to the notion of a matrix-valued holomorphic
quantum modular form. This is a tale of several independent discoveries, both
empirical and theoretical, all illustrated by the three simplest hyperbolic
knots.
