The ADO Invariants are a q-Holonomic Family

TL;DR

This paper proves that ADO link invariants form a q-holonomic family with recursion relations independent of the parameter r, linking them to the colored Jones polynomials and supporting a physics-inspired conjecture.

Contribution

It establishes the q-holonomic nature of ADO invariants and connects their recursion ideals to those of colored Jones polynomials, confirming a conjecture.

Findings

ADO invariants are q-holonomic for r ≥ 2.

The recursion ideal of ADO invariants is contained in that of colored Jones polynomials.

This confirms a physics-motivated conjecture relating ADO and Jones invariants.

Abstract

We investigate the $q$-holonomic properties of a class of link invariants based on quantum group representations with vanishing quantum dimensions, motivated by the search for the invariants' realization in physics. Some of the best known invariants of this type, constructed from `typical' representations of the unrolled quantum group $\mathcal U^H_{\zeta_{2r}}(\mathfrak{sl}_2)$ at a $2r$-th root of unity, were introduced by Akutsu-Deguchi-Ohtsuki (ADO). We prove that the ADO invariants for $r\geq 2$ are a $q$-holonomic family, implying in particular that they satisfy recursion relations that are independent of $r$. In the case of a knot, we prove that the $q$-holonomic recursion ideal of the ADO invariants is contained in the recursion ideal of the colored Jones polynomials, the subject of the celebrated AJ Conjecture. (Combined with a recent result of S. Willetts, this establishes an isomorphism of the ADO and Jones recursion ideals. Our results also confirm a recent physically-motivated conjecture of Gukov-Hsin-Nakajima-Park-Pei-Sopenko.)