ProofAtlas

Quantum topology and hyperbolic knot theory

Kashaev–Murakami–Murakami Volume Conjecture

Collaboration beta

The conjecture predicts that quantum knot invariants grow at a rate set by the hyperbolic volume of the knot complement. The packet builds exact odd-level and boundary-state machinery, but still lacks universal asymptotics and an even-level geometric state sum, so the full all-integer limit is open.

limN→∞2πNlog|JN(K;e2πi/N)|=Vol(S3∖K)
Known results and sources
A cream knot hovers over a gold hyperbolic tetrahedral volume, encircled by quantum marks with rust-colored breaks in the even sectors.
The conjecture links quantum-invariant growth to hyperbolic volume; odd-level structure is known, but the all-level bridge remains open.

Research problem

Exact mathematical statement

For every hyperbolic knot K⊂S3K\subset S^3, with qN=e2πi/Nq_N=e^{2\pi i/N} and the normalization JN(U;q)=1J_N(U;q)=1, the conjecture is

limN→∞2πNlog|JN(K;qN)|=Vol(S3∖K).\lim_{N\to\infty}\frac{2\pi}{N}\log\left|J_N(K;q_N)\right|=\operatorname{Vol}(S^3\setminus K).

The Murakami–Murakami extension replaces the right side for an arbitrary knot by the sum of the volumes of its hyperbolic JSJ pieces.

Problem infographic

Problem at a glance

A wide diagram connects a knot and tetrahedral volume to dense quantum state rings, one Fourier-selected boundary ray, a saddle landscape, and a broken even-level tensor seam.
The source combines an odd-level invariant comparison with a boundary-state route, while universal saddle control and even-level tetrahedral data remain open.

Current mathematical picture

Where work on Kashaev–Murakami–Murakami Volume Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureOdd-subsequence promotion

The source states that changing N to N plus or minus one changes both color and root, with no subexponential comparison available. Use the exact odd-level QH pair route for a correctly labeled odd-subsequence theorem while building an independent even-level exit.

Route status · Narrowed route
Main reductionFixed-meridian selection

In a meridian-adapted finite Fourier basis, the source says filling selects one peripheral residue exactly.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeAudit the source-locked BB and MM tensor formulas into one exact basis and normalization convention.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Kashaev–Murakami–Murakami Volume Conjecture in numbers

1.4kretained lines of mathematical investigation1,447 in the current working snapshot
Argument development
1,134 · 78%
Explored or eliminated routes
32 · 2%
Computational analysis
45 · 3%
Open obligations
99 · 7%
Definitions and setup
137 · 9%
7selected mapped statements2routes investigated5open questions5contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byRinat M. Kashaev, Hitoshi Murakami, Jun MurakamiProposed1997Status checkedAug 14, 2026
Source recordedAug 14, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

14 selected steps

Scroll horizontally to explore the route

Working route overview for Kashaev–Murakami–Murakami Volume ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Quantum growth rate equals hyperbolic knot-complement volume. — Depends on missing premiseQuantum growth rate equalshyperbolic knot-complementvolume.Current reduction — Depends on missing premiseCurrent reductionFixed-meridian selection — Depends on missing premiseFixed-meridian selectionOdd-level pair invariant — Depends on missing premiseOdd-level pair invariantClosing target — Depends on missing premiseClosing targetConditional boundary defect bound — Depends on missing premiseConditional boundary defectboundOdd subsequence limitation — Depends on missing premiseOdd subsequence limitationOdd-subsequence promotion — stoppedOdd-subsequence promotionNearby-root Fourier-orbit shortcut — stoppedNearby-root Fourier-orbitshortcutAudit the source-locked BB and MM tensor formulas into one exact basis and normalization convention. — OpenAudit the source-locked BBand MM tensor formulas intoone…Compute the saturated reduced boundary lattice and deck character for the figure-eight triangulation exactly. — OpenCompute the saturatedreduced boundary lattice anddeck…Close the universal geometric asymptotic and supply an all-level exit for even integers. — OpenClose the universalgeometric asymptotic andsupply…Universal saddle control — OpenUniversal saddle controlEven-level tetrahedral tensor — OpenEven-level tetrahedraltensor
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded
Narrowed routeOdd-subsequence promotion

The source states that changing N to N plus or minus one changes both color and root, with no subexponential comparison available. Use the exact odd-level QH pair route for a correctly labeled odd-subsequence theorem while building an independent even-level exit.

Route status · Narrowed route
Narrowed routeNearby-root Fourier-orbit shortcut

The source states that shifts of order N change cyclic-dilogarithm magnitudes exponentially. Analyze the full orbit identity or use exact Fourier residue selection in a meridian-adapted boundary basis.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Audit the source-locked BB and MM tensor formulas into one exact basis and normalization convention.Suggested move: Write an entrywise conversion worksheet for R versus check-R, indices, mirror, framing, root, color, and enhancement.
Ready to work on
02
Compute the saturated reduced boundary lattice and deck character for the figure-eight triangulation exactly.Suggested move: Produce integral matrices, saturation witnesses, induced pairing, cusp vectors, Smith form, and mod-two character certificates.
Ready to work on
03
Universal saddle control

A universal normal form, nonzero geometric coefficient, and strict gap against every competing contribution remain unproved.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Even-level tetrahedral tensor

The existing all-level Weyl operator does not construct the even-level charged tetrahedral matrix dilogarithm needed for a geometric state sum.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Close the universal geometric asymptotic and supply an all-level exit for even integers.Suggested move: Derive a fixed-meridian normal form with nonvanishing and global-gap estimates while constructing the charged metaplectic tensor.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The universal all-knot colored-Jones volume conjecture remains open. Primary and current sources support exact comparisons, proved model knots, generalized special cases, and extensive computation, but not a recognized proof for every hyperbolic knot and every sufficiently large integer level.

[3][4][5]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. PreprintA recent preprint proves a generalized asymptotic statement for the figure-eight knot over specified complex-parameter regimes, a scoped special-case advance.[4]
  2. Authoritative summaryMurakami and Yokota published a monograph presenting the conjecture, the figure-eight proof model, and generalized formulations.[3]
  3. Peer reviewedBaseilhac and Benedetti proved a comparison between Kashaev and quantum-hyperbolic link invariants with odd-level scope and stated phase conventions.[2]
  4. PreprintMurakami and Murakami related Kashaev invariants to normalized colored Jones polynomials and formulated the simplicial-volume extension.[1]
5 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusKashaev–Murakami–Murakami Volume Conjecture
Solved special caseFigure-eight knot volume conjecture

The figure-eight knot is a principal proved model, with recent work extending the parameter regime for a generalized conjecture.

[3][4]
Stronger or generalized formMurakami–Murakami simplicial-volume extension

The simplicial-volume version extends the hyperbolic statement to arbitrary knots using the hyperbolic pieces of the complement.

[1]
Related problemColored Jones volume prediction and improved convergence conjectures

Recent computational and machine-learning work explores volume prediction and modified convergence statements; it is evidence and conjecture formation, not a universal proof.

[5]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • dataset · source linked; not reproduced by ProofAtlasColored Jones data for hyperbolic knots through 15 crossings

    The preprint reports computed two- and three-colored Jones data used for prediction experiments; this does not certify the asymptotic conjecture.

    [5]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA formal library for colored Jones polynomials at roots of unity with fixed normalization
  • Formalization targetFormal hyperbolic knot-complement volume and JSJ infrastructure
  • Formalization targetMachine-checked quantum-dilogarithm state sums, tensor conventions, and asymptotic estimates
  • Formalization targetAn all-level formal bridge that treats even as well as odd levels

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

5 standing statements2 proposed statements5 open questions2 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma2 of 72
  • equivalence1 of 71
  • negative result1 of 71
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.23 displayed rows · 2 routes included
  • retained route statementQuantum growth rate equals hyperbolic knot-complement volume.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementOdd-level pair invariantintermediate
  • retained route statementFixed-meridian selectionintermediate
  • retained route statementConditional boundary defect boundintermediate
  • retained route statementOdd subsequence limitationintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureOdd-subsequence promotionreported failure
  • Useful failureNearby-root Fourier-orbit shortcutreported failure
  • Research targetAudit the source-locked BB and MM tensor formulas into one exact basis and normalization convention.open
  • Research targetCompute the saturated reduced boundary lattice and deck character for the figure-eight triangulation exactly.open
  • Research targetClose the universal geometric asymptotic and supply an all-level exit for even integers.open
  • Research targetUniversal saddle controlopen
  • Research targetEven-level tetrahedral tensoropen
  • ComputationThe current work records numerical figure-eight leading-ratio tests and proposes an exact integer reduced-lattice workflow, but no submitted attachment was executed by ProofAtlas.Source-reported computations support examples and protocol design only; they do not establish the universal asymptotic theorem. · reported unreproduced
  • Narrowed routeOdd-subsequence promotionThe source states that changing N to N plus or minus one changes both color and root, with no subexponential comparison available. Use the exact odd-level QH pair route for a correctly labeled odd-subsequence theorem while building an independent even-level exit.
  • Narrowed routeNearby-root Fourier-orbit shortcutThe source states that shifts of order N change cyclic-dilogarithm magnitudes exponentially. Analyze the full orbit identity or use exact Fourier residue selection in a meridian-adapted boundary basis.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeAudit the source-locked BB and MM tensor formulas into one exact basis and normalization convention.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

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Prepared starting pointAudit the source-locked BB and MM tensor formulas into one exact basis and normalization convention.

Kashaev–Murakami–Murakami Volume Conjecture · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

The conjecture predicts that quantum knot invariants grow at a rate set by the hyperbolic volume of the knot complement. The current work builds exact odd-level and boundary-state machinery, but still lacks universal asymptotics and an even-level geometric state sum, so the full all-integer limit is open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

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Proof attempt or partial resultSupporting notes or data
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Sources and references5 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    The colored Jones polynomials and the simplicial volume of a knotoriginal source · Hitoshi Murakami, Jun Murakami · arXiv; later Acta Mathematica · 1999-05-12 · ARXIV math/9905075 · accessed Aug 14, 2026
  2. 2
    The Kashaev and quantum hyperbolic link invariantspeer reviewed result · Stéphane Baseilhac, Riccardo Benedetti · Journal of Gökova Geometry Topology · 2011-01-10 · ARXIV 1101.1851 · accessed Aug 14, 2026
  3. 3
    Volume Conjecture for Knotssurvey or monograph · Hitoshi Murakami, Yoshiyuki Yokota · Springer · 2018-08-15 · DOI 10.1007/978-981-13-1150-5 · accessed Aug 14, 2026
  4. 4
    The generalized volume conjecture for the figure-eight knot parametrized by a complex number with small imaginary partpreprint · Hitoshi Murakami · arXiv · 2026-02-01 · ARXIV 2602.01049 · accessed Aug 14, 2026
  5. 5
    Colored Jones Polynomials and the Volume Conjecturepreprint · Mark Hughes, Vishnu Jejjala, P. Ramadevi, Pratik Roy, Vivek Kumar Singh · arXiv · 2025-02-25 · ARXIV 2502.18575 · accessed Aug 14, 2026

Important qualifications

  • Scoped primary and publisher-hosted pass, not a complete bibliography of verified knot families.
  • Generalized, deformed, and special-knot volume conjectures have different statements and are not merged.
  • Root, mirror, framing, and odd/even conventions require exact statement-level comparison.
  • No submitted packet URL was fetched.

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