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Quantum topology · hyperbolic geometry · knot invariants

Kashaev–Murakami–Murakami Volume Conjecture

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Does the exponential growth rate of a knot's colored Jones polynomial at roots of unity recover the hyperbolic or simplicial volume of the knot complement?

limN→∞2πNlog|JN(K;e2πi/N)| ?=Vol(S3∖K)
Known results and sources
A luminous hyperbolic knot complement and ideal tetrahedral volume sit opposite a root-of-unity colored Jones amplitude spiral, joined only by an unresolved visual correspondence.
The open conjecture asks whether root-of-unity colored-Jones growth records the hyperbolic volume of the knot complement.

Research problem

Exact mathematical statement

Let K⊂S3K\subset S^3 be a finite-volume hyperbolic knot, let qN=e2πi/Nq_N=e^{2\pi i/N}, and normalize the NN-colored Jones polynomial so that the unknot has value 11. The hyperbolic volume conjecture asks whether

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).

For an arbitrary knot, the Kashaev–Murakami–Murakami form replaces the right-hand side by v3|S3∖K|v_3\,\|S^3\setminus K\|, equivalently the sum of the volumes of the hyperbolic JSJ pieces. The governing source attacks the one-cusped hyperbolic prime-knot case first and says the conjecture remains unproved.

Problem infographic

Problem at a glance

A problem-first Volume Conjecture diagram states q_N=e^(2πi/N), the normalization J_N of the unknot equals 1, the hyperbolic-knot asymptotic equality, and the arbitrary-knot simplicial-volume right side, all marked open.
With the N-colored Jones polynomial normalized by J_N(unknot)=1, the conjecture compares its root-of-unity exponential growth with hyperbolic volume, or with simplicial volume for an arbitrary knot.

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 failureTreating abstract root-regular identities or an on-shell Casson–Rivin gap as complete transport and domination

Abstract root-regular pentagon and inversion identities, or a constrained Casson–Rivin gap at the complete point, do not by themselves prove convention-exact transport of the colored-Jones network or uniform domination on off-shell torus phases. For one genuinely multi-tetrahedron knot, provide a complete convention ledger and exact move word to a Laurent-polynomial geometric network, then prove a chamberwise finite-N bound whose loss includes off-shell constraint terms and whose only zero-loss points are complete logarithmic lifts with nonzero combined amplitude.

Route status · Narrowed route
Main reductionCurrent reduction

Rewrite a fixed reduced colored-Jones crossing network using pole-free finite root-of-unity identities, contract eligible forests to obtain a fixed-dimensional Laurent-polynomial cycle core, shift its coefficient torus through the complete hyperbolic point, and prove a finite-chamber exponential loss away from the complete lifts. A nonzero full-sequence complex Gaussian would then yield the desired volume growth.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeEmbed the actual reduced colored-Jones crossings into the finite root-regular calculus and give an exact move word.Task status · Ready to work on
Research-record correctionResearch-record correction

We 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.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Kashaev–Murakami–Murakami Volume Conjecture in numbers

989retained lines of mathematical investigation989 in the current working snapshot
Argument development
812 · 82%
Explored or eliminated routes
25 · 3%
Computational analysis
42 · 4%
Open obligations
44 · 4%
Definitions and setup
66 · 7%
5selected mapped statements1routes investigated7open questions7contribution-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 Kashaev, Hitoshi Murakami and Jun Murakami (colored-Jones restatement and arbitrary-knot strengthening)Proposed1996Status 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

13 selected steps

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

13 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.Does the exponential growth rate of a knot's colored Jones polynomial at roots of unity recover the hyperbolic or simplicial volume of the knot complement? — Depends on missing premiseDoes the exponential growthrate of a knot's coloredJones…Current reduction — Depends on missing premiseCurrent reductionClosing target — Depends on missing premiseClosing targetEligible ordered trees contract exactly — Depends on missing premiseEligible ordered treescontract exactlyFinite nilpotent identities are pole-free — Depends on missing premiseFinite nilpotent identitiesare pole-freeTreating abstract root-regular identities or an on-shell Casson–Rivin gap as complete transport and domination — stoppedTreating abstractroot-regular identities oran…Embed the actual reduced colored-Jones crossings into the finite root-regular calculus and give an exact move word. — OpenEmbed the actual reducedcolored-Jones crossings intothe…Prove uniform chamberwise finite-N domination with the correct off-shell constraint correction and zero-loss locus. — OpenProve uniform chamberwisefinite-N domination with thecorrect…Establish the full-sequence local complex saddle expansion and rule out cancellation among maximal sectors. — OpenEstablish the full-sequencelocal complex saddleexpansion…Quantum growth versus geometric volume remains open — OpenQuantum growth versusgeometric volume remainsopenActual crossings still need exact embedding — OpenActual crossings still needexact embeddingGlobal domination needs off-shell repair — OpenGlobal domination needsoff-shell repairMaximal saddles must not cancel — OpenMaximal saddles must notcancel
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

1 recorded
Narrowed routeTreating abstract root-regular identities or an on-shell Casson–Rivin gap as complete transport and domination

Abstract root-regular pentagon and inversion identities, or a constrained Casson–Rivin gap at the complete point, do not by themselves prove convention-exact transport of the colored-Jones network or uniform domination on off-shell torus phases. For one genuinely multi-tetrahedron knot, provide a complete convention ledger and exact move word to a Laurent-polynomial geometric network, then prove a chamberwise finite-N bound whose loss includes off-shell constraint terms and whose only zero-loss points are complete logarithmic lifts with nonzero combined amplitude.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

7 featured tasks
01
Embed the actual reduced colored-Jones crossings into the finite root-regular calculus and give an exact move word.Suggested move: Freeze one crossing, framing, cup/cap, cut-strand, and root convention for a multi-tetrahedron knot, then audit every tensor factor, Cartan power, scalar, denominator, and boundary conjugation through a finite sequence of allowed pentagon, inversion, Vandermonde, Lucas, and monomial-reordering moves.
Ready to work on
02
Prove uniform chamberwise finite-N domination with the correct off-shell constraint correction and zero-loss locus.Suggested move: Compute the integral map from presentation cycle labels to geometric angle coordinates, decompose the shifted torus into finitely many branch and endpoint chambers, and derive an explicit nonnegative loss in each chamber that includes the Lagrange-multiplier or constraint-violation term whenever the phase image leaves the affine angle slice.
Ready to work on
03
Quantum growth versus geometric volume remains open

The exact target equates normalized root-of-unity colored-Jones growth with hyperbolic volume, and the source explicitly says it is not proved.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Actual crossings still need exact embedding

The abstract finite calculus has not yet been matched to the actual reduced colored-Jones crossing tensors with all convention and boundary factors.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Global domination needs off-shell repair

The preferred volume-loss bound requires either proof that each torus chamber stays in the affine angle slice or an explicit multiplier or constraint penalty away from it.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
06
Maximal saddles must not cancel

A local WKB calculation must prove that the sum of complete-lift leading coefficients is nonzero for every sufficiently large N across all parity classes.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
07
Establish the full-sequence local complex saddle expansion and rule out cancellation among maximal sectors.Suggested move: At every complete logarithmic lift, fix gauge and meridian coordinates, calculate the reduced Hessian and determinant branch, enumerate parity and Lucas sectors, and prove that the combined leading coefficient stays nonzero for every sufficiently large N rather than only on a subsequence.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The general Kashaev–Murakami–Murakami volume conjecture remains open. Primary and official sources establish the historical formulation and important special cases, including the figure-eight knot and torus knots under their relevant normalizations, but do not furnish a convention-uniform proof for arbitrary hyperbolic knots or the full simplicial-volume extension.

[1][2][4]
External progress

What the literature has established

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

  1. Authoritative summaryMurakami and Yokota's monograph summarized the conjecture's status and gave an elementary proof for the figure-eight knot while presenting the general hyperbolic argument only as an idea of proof.[4]
  2. Historical sourceHitoshi and Jun Murakami related Kashaev's invariant to the N-colored Jones polynomial at an Nth root of unity and proposed the stronger arbitrary-knot simplicial-volume formulation.[2]
  3. PreprintKashaev and Tirkkonen proved the volume conjecture for torus knots, whose simplicial volume vanishes.[3]
  4. Historical sourceKashaev argued from particular examples that the exponential growth rate of his quantum-dilogarithm link invariant is the hyperbolic volume of the complement.[1]
4 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 has a direct elementary proof in the cited monograph. One explicit hyperbolic knot does not supply a uniform theorem for all hyperbolic knots.

[4]
Solved special caseTorus-knot volume conjecture

The torus-knot case verifies the zero-simplicial-volume behavior for a nonhyperbolic family, with a different geometric regime from finite-volume hyperbolic knot complements.

[3]
Stronger or generalized formArbitrary-knot simplicial-volume extension

The arbitrary-knot formulation replaces hyperbolic volume by simplicial volume, equivalently the total volume of the hyperbolic JSJ pieces.

[2]

Formalization opportunities

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

  • Formalization targetA statement-aligned formal definition of the normalized colored Jones polynomial at roots of unity and the exact Kashaev-invariant convention, including mirror and framing factors.
  • Formalization targetFormal hyperbolic knot-complement geometry, simplicial volume, ideal-tetrahedron volume, and JSJ decomposition with the normalization used in the conjecture.
  • Formalization targetMachine-checked asymptotic interfaces for root-of-unity q-products, exact tensor-network transformations, multidimensional stationary phase, endpoint control, and noncancellation.

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 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.

3 standing statements2 proposed statements7 open questions1 narrowed routes
Statements by mathematical role5 selected mapped statements
  • theorem candidate1 of 51
  • reduction1 of 51
  • lemma3 of 53
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.19 displayed rows · 1 route included
  • retained route statementDoes the exponential growth rate of a knot's colored Jones polynomial at roots of unity recover the hyperbolic or simplicial volume of the knot complement?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementFinite nilpotent identities are pole-freeintermediate
  • retained route statementEligible ordered trees contract exactlyintermediate
  • 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
  • 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 failureTreating abstract root-regular identities or an on-shell Casson–Rivin gap as complete transport and dominationreported failure
  • Research targetEmbed the actual reduced colored-Jones crossings into the finite root-regular calculus and give an exact move word.open
  • Research targetProve uniform chamberwise finite-N domination with the correct off-shell constraint correction and zero-loss locus.open
  • Research targetEstablish the full-sequence local complex saddle expansion and rule out cancellation among maximal sectors.open
  • Research targetQuantum growth versus geometric volume remains openopen
  • Research targetActual crossings still need exact embeddingopen
  • Research targetGlobal domination needs off-shell repairopen
  • Research targetMaximal saddles must not cancelopen
  • ComputationSource-reported small-dimensional and small-capacity sanity checks for truncated representations, ordered q-Vandermonde, root-of-unity q-Lucas, and finite nilpotent inversion and pentagon identities.The source reports roundoff-scale residuals for N=3,4,5 and other small-capacity checks, but includes no executable verification scripts. ProofAtlas did not independently reproduce them, and they are not proof of knot-network transport or the volume conjecture. · reported unreproduced
  • Narrowed routeTreating abstract root-regular identities or an on-shell Casson–Rivin gap as complete transport and dominationAbstract root-regular pentagon and inversion identities, or a constrained Casson–Rivin gap at the complete point, do not by themselves prove convention-exact transport of the colored-Jones network or uniform domination on off-shell torus phases. For one genuinely multi-tetrahedron knot, provide a complete convention ledger and exact move word to a Laurent-polynomial geometric network, then prove a chamberwise finite-N bound whose loss includes off-shell constraint terms and whose only zero-loss points are complete logarithmic lifts with nonzero combined amplitude.
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 bridgeEmbed the actual reduced colored-Jones crossings into the finite root-regular calculus and give an exact move word.

1 approach has 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.

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Prepared starting pointEmbed the actual reduced colored-Jones crossings into the finite root-regular calculus and give an exact move word.

Kashaev–Murakami–Murakami Volume Conjecture · ready to start

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Research contextPrepared context for any AI agent

Does the exponential growth rate of a knot's colored Jones polynomial at roots of unity recover the hyperbolic or simplicial volume of the knot complement?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 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 hyperbolic volume of knots from quantum dilogarithmoriginal source · R. M. Kashaev · arXiv; Letters in Mathematical Physics 39 (1997) · 1996-03-25 · ARXIV q-alg/9601025 · DOI 10.1023/A:1007364912784 · accessed Aug 14, 2026
  2. 2
    The colored Jones polynomials and the simplicial volume of a knotoriginal source · Hitoshi Murakami, Jun Murakami · arXiv; Acta Mathematica 186 (2001) · 1999-06-10 · ARXIV math/9905075 · DOI 10.1007/BF02392716 · accessed Aug 14, 2026
  3. 3
    Proof of the volume conjecture for torus knotspreprint · R. M. Kashaev, O. Tirkkonen · arXiv · 1999-12-27 · ARXIV math/9912210 · accessed Aug 14, 2026
  4. 4
    Volume Conjecture for Knotssurvey or monograph · Hitoshi Murakami, Yoshiyuki Yokota · Springer · 2018 · DOI 10.1007/978-981-13-1150-5 · accessed Aug 14, 2026

Important qualifications

  • The hyperbolic-knot volume conjecture and the arbitrary-knot simplicial-volume extension are recorded separately even though they share historical sources.
  • Special cases for the figure-eight knot and torus knots do not establish the general hyperbolic or arbitrary-knot statement.
  • The scoped source review did not identify a primary source proving the submitted packet's root-regular transport and chamberwise domination theorem for arbitrary hyperbolic knots; this does not prove no such development exists.
  • No submitted URL was fetched, and the current work's internal small-N checks were not treated as independently reproducible evidence.
  • No statement-aligned formalization, checked proof certificate, reusable dataset, or independently reproduced computation was identified in this collection.

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