Arithmetic Volume Conjecture for Kashaev Invariants

Definitions and mathematical statement

Source-level question. Does the Volume Conjecture hold for Kashaev invariants, including an arithmetic refinement relating their root-of-unity asymptotics to the knot trace field and modular transformations?

Definitions and unresolved qualifications. Write $K$ for a knot and $\langle K\rangle_N$ for the invariant indexed by a positive integer $N$ that the record calls a Kashaev invariant. This notation is only a placeholder: the supplied record does not define the invariant, its normalization, its framing convention, or its root-of-unity convention. It also does not specify the ambient manifold, the class of knots being quantified over, the volume appearing in the conjecture, or the asymptotic identity to be proved.

The arithmetic refinement is likewise unspecified. The record supplies no formula, limiting regime, asymptotic coefficients, field-membership assertion, or modular transformation law. It does not define the intended knot trace field or distinguish it from an invariant trace field. Consequently, the supplied question cannot be converted into a uniquely determined, fully quantified mathematical conjecture without additional source material. The volume component and the arithmetic refinement both remain requested targets; the latter cannot be replaced by the volume component alone.

Short English statement

The record asks whether Kashaev invariants satisfy the Volume Conjecture together with an arithmetic refinement involving roots of unity, a knot trace field, and modular transformations. Essential definitions and formulas are absent, so the precise conjecture requires clarification from the cited source. The supplied status is partially_solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry, permanent ID 30001535, source problem code catalogue entry, title Arithmetic Volume Conjecture for Kashaev Invariants, in the Oberwolfach Reports Open Problems collection. No individual proposer is named; the supplied proposed year is 2010.
• Supplied report citation: Low-Dimensional Topology and Number Theory (2011), Oberwolfach Reports, DOI 10.4171/owr/2010/35, pp. 2101--2163. https://doi.org/10.4171/owr/2010/35. This is also the supplied literature source labeled "original OWR report." The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No pinpoint passage containing the conjecture is supplied.
• Supplied literature source labeled "root-of-unity asymptotics": https://link.springer.com/article/10.1007/s11139-020-00266-x. No authors, article title, theorem number, or supporting excerpt is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports substantial proofs and asymptotic results in special classes, while describing the general knot-level statement in the extracted passage as unresolved. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.