Problem 1.22 — (H.

Definitions and mathematical statement

A torus knot $K=T(p,q)\subset S^3$ is a simple closed curve of primitive slope $(p,q)$ on a standard embedded torus, with $p,q$ coprime. The source asks to calculate a suitably defined Chern--Simons quantity $\operatorname{CS}(S^3\setminus K)$ and a suitably branched logarithmic colored-Jones limit, extracted as \[ \lim_{N\to\infty}\frac{\operatorname{Log} J_N(K)}{N}. \] Giving the appropriate Chern--Simons definition and choosing the logarithm branch are expressly parts of the task.

For a knot complement, a Chern--Simons quantity generally requires a connection/representation, gauge group, normalization and boundary/peripheral data. The open complement and the compact exterior $S^3\setminus\operatorname{int}\nu K$ require compatible boundary treatment. No canonical hyperbolic structure is assigned to a torus-knot complement.

The usual colored Jones polynomial $J_N(K;q)$ needs a color convention and unknot normalization. The excerpt omits the evaluation defining the numerical $J_N(K)$. A root such as $q=e^{2\pi i/N}$ is a familiar possible context, but no specialization is inserted without the preceding definitions. Nonzero values and a coherent logarithm choice are required: changing branches by $2\pi i k_N$ can change the normalized limit. Knot orientation/mirror, torus parameters and framing conventions must also be fixed.

The registry labels this solved, while the summary says "Classification: PARTIAL-PROGRESS", reports some Chern--Simons computations and an understood Jones limit, and gives no formulas or exact theorem references. The discrepancy remains unresolved rather than being treated as verification of a complete all-torus-knot answer.

Short English statement

For torus knots, define and compute the relevant Chern--Simons quantity and normalized logarithmic colored-Jones limit with an appropriate branch. The numerical specialization, boundary data and normalizations are missing, and the supplied solved/partial status claims conflict.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400022$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 1.22, attributed to H. Murakami, PDF page 25: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29; provenance distinguishes 231 numbered items from surrounding references.
• No separate Chern--Simons formula, Jones-asymptotic theorem or resolving publication is supplied.