Problem 7.13 — (H.

Definitions and mathematical statement

A Seifert-fibered $3$-manifold is a manifold decomposed into circle fibers with standard fibered solid-torus local models, allowing exceptional fibers; equivalently its fibration has a $2$-orbifold base in the usual setting. Let $\tau_N^{\mathrm{SU}(2)}(M)$ be the source's $\mathrm{SU}(2)$ quantum invariant at index $N$. The normalization, root of unity, admissible indices and manifold domain of this invariant are not defined in the supplied extract.

H. Murakami's Problem 7.13 asks to calculate \[ \operatorname{o\!\!-lim}_{N\to\infty} \frac{\log\tau_N^{\mathrm{SU}(2)}(M)}{N} \] for Seifert-fibered $3$-manifolds $M$. The notation is the source's optimistic limit, not an ordinary limit asserted to exist. Its defining prescription is absent: the record does not state how a quantum-invariant expression is transformed, which critical values or branches are used, or whether the resulting quantity is canonical. It also gives no logarithm-branch convention or treatment of zero values. Thus neither a genuine complex limit nor a real $\log|\tau_N|$ limit may be substituted for this expression. A complete formulation must recover the optimistic-limit procedure and the allowed Seifert data and index sequences before an explicit value can be requested unambiguously.

Short English statement

Calculate the optimistic logarithmic large-index asymptotic of the $\mathrm{SU}(2)$ quantum invariant for Seifert-fibered three-manifolds. The optimistic-limit prescription is missing and must be distinguished from ordinary convergence. The supplied partial-progress assessment says some computations exist but does not cite them.

Sources

• Local UnsolvedMath registry, permanent ID $10400128$, exact source code catalogue entry, Problem 7.13, attributed to H. Murakami. The supplied title is truncated after "(H."
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 7.13, PDF page 110: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Bold printed headings distinguish 231 numbered items from surrounding references.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Registry status partially_solved and classification PARTIAL-PROGRESS report some optimistic-limit computations for Seifert-fibered spaces, without specific primary references.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.