The descendant colored Jones polynomials

Abstract

We discuss two realizations of the colored Jones polynomials of a knot, one from an unnoticed work of the second author in 1994 on quantum R-matrices at roots of unity obtained from solutions of the pentagon identity, and another one from recent work of D. Zagier and the first author regarding the Refined Quantum Modularity Conjecture, more precisely, the mysterious top row of a matrix of conjectured knot invariants.