Generic Uniqueness for Skein-Algebra Representations

Definitions and mathematical statement

Let $N$ be a positive odd integer, and let $A\in{\mathbb{C}}$ have multiplicative order $2N$. Let $S$ be a surface satisfying the hypotheses intended in the source, and write $\mathcal S_A(S)$ for its Kauffman bracket skein algebra over ${\mathbb{C}}$. The supplied record does not specify the class of surfaces, their boundary conditions, or the skein conventions. These remain qualifications on the formulation below.

Write \[ X(S)=\operatorname{Hom}(\pi_1(S),\mathrm{SL}_2({\mathbb{C}}))\mathbin{/\!/}\mathrm{SL}_2({\mathbb{C}}) \] for the character variety, with simultaneous conjugation and the affine quotient interpretation under the intended surface hypotheses. An irreducible finite-dimensional representation is a unital algebra homomorphism $\rho:\mathcal S_A(S)\to\operatorname{End}_{{\mathbb{C}}}(V)$, where $0<\dim_{{\mathbb{C}}}V<\infty$ and $V$ has no nonzero proper invariant subspace. Two representations $\rho$ and $\rho'$ are equivalent if there is a linear isomorphism $T:V\to V'$ such that $T\rho(x)=\rho'(x)T$ for every $x\in\mathcal S_A(S)$.

For each representation to which the source correspondence applies, denote its classical shadow by $\operatorname{sh}(\rho)\in X(S)$, and denote its collection of puncture invariants by $\mathbf p(\rho)$. Here these are names for the data in the source's Theorems 1 and 2: their defining identities, normalization, puncture index set, and compatibility conditions are absent from the supplied record and are not stipulated by this draft.

Generic uniqueness question. For every surface and root parameter covered by the intended source hypotheses, does there exist a Zariski-open subset $U\subseteq X(S)$, with $\overline U^{\mathrm{Zar}}=X(S)$, such that for every pair of irreducible finite-dimensional representations in the source correspondence, \[ \operatorname{sh}(\rho)=\operatorname{sh}(\rho')\in U, \qquad \mathbf p(\rho)=\mathbf p(\rho') \quad\Longrightarrow\quad \rho\cong\rho'? \] The puncture data are required to satisfy the source's compatibility conditions with the common classical shadow. This implication expresses the uniqueness requested by the supplied clean statement. The original question asks whether the correspondence in Theorems 1 and 2 is one-to-one over a Zariski-dense open subset; identifying its precise domain and codomain, and any accompanying existence assertion for compatible data, requires the omitted theorem statements. Consequently this is a conditional statement-only formulation requiring mathematical review.

Short English statement

For the surface class intended by the source, and a primitive $2N$-th root $A$ with $N$ odd, ask whether representations with generic classical shadow are determined up to equivalence by that shadow and their compatible puncture invariants. The surface hypotheses and the definitions and compatibility rules from the referenced theorems are missing, so the precise correspondence remains unresolved.

Sources

• Local registry: data/unsolvedmath/problems.json and the supplementary registry catalog/problems.json; supplied source record with permanent UnsolvedMath ID $30002911$, title Generic Uniqueness for Skein-Algebra Representations, exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: New Perspectives on the Interplay between Discrete Groups in Low-Dimensional Topology and Arithmetic Lattices (2016), Oberwolfach Reports, pp. 1701--1746, DOI 10.4171/owr/2015/30. Source URL and literature-source label "original Oberwolfach report": https://doi.org/10.4171/owr/2015/30. The original question refers to Theorems 1 and 2, whose statements are not included in the supplied record.
• Supplied literature source labelled "Bonahon and Liu primary paper": https://arxiv.org/abs/math/0407086. No title, theorem excerpt, or explicit connection to the stated correspondence is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the generic one-to-one correspondence remains conjectural and that the cited classification results do not prove it. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report passage was checked directly and its question, definitions, and notation restored without strengthening the claim. These are supplied provenance assertions; this draft does not independently verify them or claim to have visited the sources.