[#P3114] Kashaev volume conjecture for hyperbolic knots

Quantum knot invariants approaching hyperbolic volume. — A structural knot diagram of the statement's mathematical objects.

Problem. For every hyperbolic knot \(K\subset S^3\), does \(\lim_{N\to\infty}(2\pi/N)\log|\langle K\rangle_N|=\operatorname{Vol}(S^3\setminus K)\), where \(\langle K\rangle_N\) is Kashaev's \(N\)-th quantum invariant?

1Context

Known frontier: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. Open boundary: A proof or counterexample for arbitrary hyperbolic knots remains unknown.

2Problem setup

Definition 1 (hyperbolic knot). A knot whose complement admits a complete finite-volume hyperbolic metric.

Definition 2 (Kashaev invariant). The Nth root-of-unity knot invariant introduced from the quantum dilogarithm.

Remark 1. The conjecture equates exponential growth of quantum knot invariants at roots of unity with the hyperbolic volume of the knot complement. The Kashaev formulation avoids normalization ambiguity in colored Jones notation.

3What counts as a solution

Prove existence of the limit and the stated equality for every hyperbolic knot.
Or give a hyperbolic knot for which the limit fails to exist or differs from hyperbolic volume.

1Status

Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. Exact unresolved remainder: A proof or counterexample for arbitrary hyperbolic knots remains unknown.^[1][2]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. Exact unresolved remainder: A proof or counterexample for arbitrary hyperbolic knots remains unknown.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Kashaev volume conjecture for hyperbolic knots. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. The exact unresolved remainder is: A proof or counterexample for arbitrary hyperbolic knots remains unknown. A complete resolution must meet the following acceptance conditions: - Prove existence of the limit and the stated equality for every hyperbolic knot. - Or give a hyperbolic knot for which the limit fails to exist or differs from hyperbolic volume. Open full record →
	Result	Supported
claim · Claim 2 The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements.^[1][2] Relevance to this problem Locates the present research frontier immediately below Kashaev volume conjecture for hyperbolic knots. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. This leaves the following boundary unresolved: A proof or counterexample for arbitrary hyperbolic knots remains unknown. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
	Trace	Supported
attempt · Route 1 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. Unresolved remainder: A proof or counterexample for arbitrary hyperbolic knots remains unknown.^[1][2] Relevance to this problem Documents why Kashaev volume conjecture for hyperbolic knots was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - Kashaev volume conjecture general hyperbolic knots open 2026 - volume conjecture proven classes knots current status The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. Unresolved remainder: A proof or counterexample for arbitrary hyperbolic knots remains unknown. Related: Claim 1 Open full record →
	Trace	Reported
attempt · Route 2 A proof or counterexample for arbitrary hyperbolic knots remains unknown. Relevance to this problem Turns the remaining uncertainty in Kashaev volume conjecture for hyperbolic knots into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: A proof or counterexample for arbitrary hyperbolic knots remains unknown. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Prove existence of the limit and the stated equality for every hyperbolic knot. - Or give a hyperbolic knot for which the limit fails to exist or differs from hyperbolic volume. Related: Claim 1 Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements. Exact unresolved remainder: A proof or counterexample for arbitrary hyperbolic knots remains unknown.

Equivalent-formulation queries: Kashaev volume conjecture general hyperbolic knots open 2026; volume conjecture proven classes knots current status
Strongest checked neighboring result: The conjecture is proved for several knot families and supported by extensive asymptotic calculations and geometric refinements.
Exact unresolved remainder: A proof or counterexample for arbitrary hyperbolic knots remains unknown.

How the 4 records connectTyped relations and evidence flow

ProblemKashaev volume conjecture for hyperbolic knots

Claim 1Current status and exact unresolved remainderin this packetSupported
Route 1Dated source and duplicate auditchecksSupported
Claim 2Strongest checked neighboring resultinformsSupported
Route 2Work at the unresolved boundaryaddressesReported

2See also

Whitehead asphericity conjecturetopology
Purely cosmetic surgery conjecturetopology
Dürer’s edge-unfolding problemtopology

How to cite

TheoremDB contributors, “Kashaev volume conjecture for hyperbolic knots,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/kashaev-volume-conjecture

This page as plain text: kashaev-volume-conjecture.md

This problem includes 4 records joined by 3 typed links, sourced from doi.org^[1], current as of August 1, 2026.

1References

Packet source. R. M. KASHAEV, “The Hyperbolic Volume of Knots from the Quantum Dilogarithm”. Letters in Mathematical Physics 39(3) (1997), 269-275. DOI 10.1023/A:1007364912784. volume-growth conjecture. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Introduces the invariant and the volume asymptotic.Also cited at R. Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Letters in Mathematical Physics 39 (1997). volume-growth conjecture.Source used to assess the problem's recorded status.For Kashaev volume conjecture for hyperbolic knots: This is the dated publication status for the canonical target Kashaev volume conjecture for hyperbolic knots.Source named by the research packet.
Hitoshi Murakami and Jun Murakami, “The colored Jones polynomials and the simplicial volume of a knot”. Acta Mathematica 186(1) (2001), 85-104. DOI 10.1007/BF02392716. Conjecture 2.2 and the examples that follow. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Identifies Kashaev's invariant with a specialization of the colored Jones polynomial and states the volume conjecture in that form.Source used to assess the problem's recorded status.For Kashaev volume conjecture for hyperbolic knots: Identifies Kashaev's invariant with a specialization of the colored Jones polynomial and states the volume conjecture in that form.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

[#P3114] Kashaev volume conjecture for hyperbolic knots

1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance