The descendant colored Jones polynomials

TL;DR

This paper explores two different approaches to understanding the colored Jones polynomials of knots, connecting historical quantum R-matrix solutions with recent conjectures on quantum modularity.

Contribution

It introduces a novel link between classical quantum R-matrix solutions and modern quantum modularity conjectures in knot theory.

Findings

Two realizations of colored Jones polynomials are presented.

Connections between quantum R-matrices and quantum modularity are established.

New insights into the structure of knot invariants are proposed.

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.