A unification of the ADO and colored Jones polynomials of a knot

Abstract

In this paper we prove that the family of colored Jones polynomials of a knot in $S^3$ determines the family of ADO polynomials of this knot. More precisely, we construct a two variables knot invariant unifying both the ADO and the colored Jones polynomials. On the one hand, the first variable $q$ can be evaluated at $2r$ roots of unity with $r\in {\mathbb{N}}^{\ast }$ and we obtain the ADO polynomial over the Alexander polynomial. On the other hand, the second variable $A$ evaluated at $A=q^n$ gives the colored Jones polynomials. From this, we exhibit a map sending, for any knot, the family of colored Jones polynomials to the family of ADO polynomials. As a direct application of this fact, we will prove that every ADO polynomial is holonomic and is annihilated by the same polynomial as of the colored Jones function. The construction of the unified invariant will use completions of rings and algebra. We will also show how to recover our invariant from Habiro’s quantum ${\mathfrak{sl}}_2$ completion studied by Habiro in [J. Pure Appl. Algebra 211 (2007), 265–292], showing that it corresponds in fact to the two-variable colored Jones invariant defined by Habiro in [Invent. Math. 171 (2008), 1–81].