Parametrization of Turaev–Viro Three-Manifold Invariants

Definitions and mathematical statement

Let ${\mathbb{Z}}_{>0}=\{1,2,3,\ldots\}$. For each $N\in{\mathbb{Z}}_{>0}$, write $\mathcal C_{N,1,0}$ for the fusion category designated by this notation in the supplied source record. Write $\operatorname{TV}_N$ for the three-manifold invariant associated with its Turaev--Viro topological quantum field theory, subject to the construction and conventions intended by that source.

Source conjecture and requested task. The original formulation conjectures that the family \[ \left(\operatorname{TV}_N\right)_{N\in{\mathbb{Z}}_{>0}} \] admits a parametrization as $N$ varies. The supplied edited formulation asks for a uniform parametrization of this family.

Unresolved qualifications. The record does not define $\mathcal C_{N,1,0}$, specify its base field or the categorical structure and normalization used for its Turaev--Viro construction, or state the admissible class of three-manifolds, including orientation and boundary conventions. It also supplies neither the proposed parameters nor a definition of what constitutes a uniform parametrization. Thus the displayed family names the intended objects but does not specify a parametrization theorem with a definite mathematical conclusion. These omissions prevent a fully precise formulation from the supplied evidence alone.

Short English statement

For every positive integer $N$, consider the Turaev--Viro three-manifold invariant associated with the source-designated fusion category $\mathcal C_{N,1,0}$. The task is to describe these invariants uniformly as $N$ varies. The categories, manifold conventions, and intended meaning of parametrization require clarification before this becomes a precise conjecture.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004284$, source problem code catalogue entry, titled "Parametrization of Turaev--Viro Three-Manifold Invariants," in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2019$ and names no individual proposer.
• Supplied report citation: Subfactors and Applications (2020), Oberwolfach Reports, pp. 3053--3140, DOI 10.4171/owr/2019/49. This is also the supplied literature source labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2019/49. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "Turaev--Viro, state-sum invariants and quantum 6j-symbols": https://doi.org/10.1016/0040-9383(92)90015-A.
• Literature assessment dated 2026-08-22: the supplied curation labels the record partially_solved. It reports that an exceptional quantum-subgroup fusion-rule question was answered, while the separate conjecture concerning the proposed parametrization of the associated Turaev--Viro invariant is not shown to be proved. This is supplied dated curation derived from the OWR record, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and states that notation, typography, and sentence structure were repaired while preserving mathematical content. This supplied editorial designation does not constitute independent mathematical verification of this draft.