The source requires signed local graphs, exact knot diagrams, legal moves, and preservation of reducedness and minimality for each topological exit. Exact arithmetic sieves remain useful within the prism locus when paired with separately certified embedded topological exits.
Route status · Narrowed routeKnot theory and quantum topology
Jones Unknot Conjecture
Collaboration betaDoes Jones polynomial 1 force a knot to be the unknot? The question remains open. It is verified through 24 crossings, while the retained source studies only a restricted weighted-prism program with explicit global gaps.
Known results and sources
Research problem
Exact mathematical statement
Let be a knot and let be its normalized Jones polynomial, with . Does
hold for every knot ?
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Jones Unknot Conjecture stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
Use a weighted triangular-prism kernel as a restricted test case, then keep isolated arithmetic, embedded topology, smoothing links, and arbitrary rigid-kernel reduction as separate obligations.
Evidence posture · Source-reported route statement · dependencies incompleteWe 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 We removed a duplicate or outdated task or route step.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 unchangedWork mapped so far
Jones Unknot Conjecture in numbers
4.3kretained lines of mathematical investigation4,344 in the current working snapshot
- Argument development
- 3,551 · 82%
- Explored or eliminated routes
- 116 · 3%
- Computational analysis
- 311 · 7%
- Open obligations
- 193 · 4%
- Definitions and setup
- 173 · 4%
7selected mapped statements2routes investigated4open questions4contribution-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.
Recommended next task
Cover all prism candidates with at least two exceptional triangle-edge residues before treating the thirteen-phase list as global.
Suggested move: Partition every omitted exceptional-residue support and record phase, orbit, determinant, and lift status.
What would count as progress
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
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
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
Scroll horizontally to explore the route
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
The source requires signed local graphs, exact knot diagrams, legal moves, and preservation of reducedness and minimality for each topological exit. Exact arithmetic sieves remain useful within the prism locus when paired with separately certified embedded topological exits.
Route status · Narrowed routeThe source lists arbitrary rigid kernels and smoothing-link topology as independent unresolved layers. The prism remains a legitimate test case and source of bounded exact subproblems, with no universal-reduction claim.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Cover all prism candidates with at least two exceptional triangle-edge residues before treating the thirteen-phase list as global.Suggested move: Partition every omitted exceptional-residue support and record phase, orbit, determinant, and lift status.
Ready to work on
Eliminate or completely certify the trivial-stabilizer isolated prism points after the exceptional-residue strata are covered.Suggested move: Stratify the symmetry-free locus by phase and mod-11 orbit, then combine determinant, moment, multi-prime, Hensel, and coordinate bounds.
Ready to work on
Exact Jones unknot target remains open
The exact conjecture asks whether V_K(t)=1 forces K to be isotopic to the unknot; the current claim ledger marks it open.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Supply embedded topology certificates, an arbitrary rigid-kernel theorem, and a noncircular smoothing-link theorem without merging their dependencies.Suggested move: Build explicit diagram-level certificates for algebraic exits and, independently, pursue arbitrary rigid-kernel and smoothing-link theorems.
Ready to work on
Sourced mathematical context
The known mathematical landscape
External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Authoritative summaryA Bulletin of the AMS overview presents the Jones polynomial in modern knot-theoretic context; the finite verification remains distinct from general detection.[3] Computational resultTuzun and Sikora verified the conjecture for every knot through 24 crossings.[2] Historical sourceJones introduced the normalized polynomial invariant with value 1 on the unknot.[1]
Mathematical neighborhood
Related results and reusable starting points
Current focusJones Unknot Conjecture
The conjecture holds for every knot with at most 24 crossings; higher-crossing and unrestricted knots remain outside that computation.
[2]Formal and computational footholds
Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.
- computation · not independently reproducedVerification through 24 crossings
The peer-reviewed result reports complete verification through 24 crossings; this collection did not rerun its enumeration.
[2]
Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetFormal knot isotopy and normalized Jones polynomial infrastructure.
- Formalization targetA formal bridge from any algebraic graph or tangle reductions to exact knot-diagram moves.
- Formalization targetA general proof that V_K(t)=1 forces K to be the unknot, beyond finite crossing enumeration.
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.
Correction detailsCorrected the research recordCorrection note
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 statements4 open questions2 narrowed routes
- theorem candidate
1 of 7 1 - reduction
1 of 7 1 - lemma
2 of 7 2 - computational claim
1 of 7 1 - negative result
2 of 7 2
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 statementDoes the Jones polynomial detect the unknot?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementRestricted prism exponent boundintermediate
- retained route statementSwapped-reflection locus is emptyintermediate
- retained route statementComposite five-symmetry exclusion with caveatintermediate
- retained route statementThirteen-phase result has restricted scopeintermediate
- 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 failureAlgebraic unit-edge promotion without embedded topologyreported failure
- Useful failureTreating a complete prism calculation as a global proofreported failure
- Research targetCover all prism candidates with at least two exceptional triangle-edge residues before treating the thirteen-phase list as global.open
- Research targetEliminate or completely certify the trivial-stabilizer isolated prism points after the exceptional-residue strata are covered.open
- Research targetSupply embedded topology certificates, an arbitrary rigid-kernel theorem, and a noncircular smoothing-link theorem without merging their dependencies.open
- Research targetExact Jones unknot target remains openopen
- Research targetGlobal topology and kernel bridges remain opensuperseded
- ComputationWithin the locally irreducible swapped-reflection prism locus, the source bounds the finite boxes and checks that every residual point violates |U|≤45.No locally irreducible swapped-reflection prism candidate exists; this conclusion is restricted to that symmetry locus. · reported unreproduced
- Narrowed routeAlgebraic unit-edge promotion without embedded topologyThe source requires signed local graphs, exact knot diagrams, legal moves, and preservation of reducedness and minimality for each topological exit. Exact arithmetic sieves remain useful within the prism locus when paired with separately certified embedded topological exits.
- Narrowed routeTreating a complete prism calculation as a global proofThe source lists arbitrary rigid kernels and smoothing-link topology as independent unresolved layers. The prism remains a legitimate test case and source of bounded exact subproblems, with no universal-reduction claim.
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
2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.
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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Prepared starting pointCover all prism candidates with at least two exceptional triangle-edge residues before treating the thirteen-phase list as global.
Jones Unknot Conjecture · ready to start
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Research contextPrepared context for any AI agent
Does Jones polynomial 1 force a knot to be the unknot? The question remains open. It is verified through 24 crossings, while the retained source studies only a restricted weighted-prism program with explicit global gaps.
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
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Sources and references3 cited works · next context review by Nov 15, 2026
The mathematical context was checked on Aug 15, 2026. Status can be refreshed sooner after a material result or claim.
- 1A polynomial invariant for knots via von Neumann algebrasoriginal source · Vaughan F. R. Jones · Bulletin of the American Mathematical Society · 1985 · DOI 10.1090/S0273-0979-1985-15304-2 · accessed Aug 15, 2026
- 2Verification of the Jones unknot conjecture up to 24 crossingspeer reviewed result · Robert E. Tuzun, Adam S. Sikora · Journal of Knot Theory and Its Ramifications · 2021-04-29 · ARXIV 2003.06724 · DOI 10.1142/S0218216521500206 · accessed Aug 15, 2026
- 3The Jones polynomial, knots, diagrams, and categoriessurvey or monograph · Bulletin of the American Mathematical Society · 2023 · DOI 10.1090/bull/1792 · accessed Aug 15, 2026
Important qualifications
- The review focuses on the original invariant, a modern AMS overview, and the strongest retained finite-crossing verification; it is not an exhaustive knot-polynomial bibliography.
- Finite verification through 24 crossings is computational evidence for a bounded class and does not prove the unrestricted conjecture.
- No end-to-end formalization or independently rerun 24-crossing computation was established in this metadata collection.
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