ProofAtlas

Topological graph theory · graph drawings · binary linear algebra

Conway’s Thrackle Conjecture

Collaboration beta

In a drawing where every pair of distinct edges meets exactly once, the conjecture says the graph cannot have more edges than vertices.

Ghas a thrackle drawing⇒|E(G)|≤|V(G)|
Conway $1,000 prize problemListed inFrontierMath: Open Problems
Known results and sources
A five-vertex pentagram thrackle uses five gold edges and five emerald vertex discs; adjacent edges meet at a vertex and nonadjacent edges cross once.
The five-cycle pentagram is an equality example with five vertices and five edges.

Research problem

Exact mathematical statement

A thrackle drawing is a graph drawing in which every pair of distinct edges meets exactly once, either at a shared endpoint or at one proper transverse crossing. Conway’s conjecture states

Ghas a thrackle drawing⇒|E(G)|≤|V(G)|.G \text{ has a thrackle drawing} \quad\Longrightarrow\quad |E(G)| \le |V(G)|.

The conjecture remains open. The results below are partial and depend on two literature statements that still need independent checking.

Problem infographic

Problem at a glance

A forest-green scientific plate centers a five-cycle pentagram with exactly five filled emerald graph vertices, five ivory edges, and five hollow gold proper-crossing markers. Two small glyphs distinguish a shared endpoint from one transverse crossing; a gold question asks whether every thrackle drawing satisfies the edge–vertex bound. A vermilion inset shows that two disjoint edges do not form a thrackle.
In a thrackle drawing, every pair of distinct edges meets exactly once, either at a shared endpoint or at one proper transverse crossing. Conway’s conjecture says every such drawing has at most as many edges as vertices. The five-cycle pentagram attains equality with five edges and five vertices.

Current mathematical picture

Where work on Conway’s Thrackle Conjecture stands

Open conjecture

Selected mathematical highlights from the fourth-audit source material: a conditional even–even figure-eight reduction, distinguished-word and exact F₂ normal forms, endpoint-parity and bicycle decompositions, exact C₆/C₈ and square-zero F₆,₆ computations, several refuted algebraic shortcuts, and the current legal two-edge block-surgery or bounded-certificate frontier. The two starting literature theorems remain unaudited here, the geometric realization bridge is open, and the thrackle conjecture is not proved.

Leading routePositive-defect localization

Use the rank-K residue budget only after it is realized by genuine block-local word features and tied to a strict descent measure.

Route status · Active route
Useful failureBranched-double-cover residues

The first two 2-adic stages yield no new obstruction; any continuation would need to encode the global crossing orders rather than only sign and magnitude data.

Route status · Eliminated route
Main reductionRank-one and higher-rank channels classified algebraically

Rank-one charged and pure interfaces receive exact local structure, while zero-defect higher rank decomposes into opposite-eigenvalue algebraic channels.

Evidence posture · Reported reduction
Completed special caseParity defect and C₆ rigidity

The endpoint-parity identity is derived, and the eight classified C₆ words admit a symbolic zero-defect rigidity proof.

Evidence posture · Reported special case
Priority open bridgeExclude every constant-parity square-zero core

Prove that no block-respecting square-zero order pair supports both complete interval systems in any of the four constant-parity sectors.

Task status · Ready to work on
Earlier research stageLegal block-surgery frontier

The first missing internal theorem is now a realizable rank-one channel plus legal two-edge block surgery, or a bounded affine certificate.

Research stage 10

Work mapped so far

Conway’s Thrackle Conjecture in numbers

2kretained lines of mathematical investigation1,988 in the current working snapshot
Argument development
1,680 · 85%
Explored or eliminated routes
62 · 3%
Computational analysis
87 · 4%
Open obligations
32 · 2%
Definitions and setup
127 · 6%
21selected mapped statements12routes investigated10reported milestones6open 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 byJohn Horton ConwayStatus checkedAug 2, 2026
Source recordedAug 2, 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

26 selected steps

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

26 selected steps

Scroll horizontally to explore the route

Working route overview for Conway’s Thrackle ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Conway’s thrackle conjecture — Depends on missing premiseConway’s thrackle conjectureConstant parity forces a square-zero core — Depends on missing premiseConstant parity forces asquare-zero coreDistinguished-word equivalence — ChallengedDistinguished-wordequivalenceEndpoint bicycles are the zero-defect ghost space — Depends on missing premiseEndpoint bicycles are thezero-defect ghost spaceEven–even figure-eight reduction — ChallengedEven–even figure-eightreductionExact F₂ normal form — Depends on missing premiseExact F₂ normal formPositive-defect bicycle purification — Depends on missing premisePositive-defect bicyclepurificationAffine obstruction and dual certificate — Depends on missing premiseAffine obstruction and dualcertificateC₆ parity rigidity — Depends on missing premiseC₆ parity rigidityExact defect hierarchy — Depends on missing premiseExact defect hierarchyExact endpoint-graph model — Depends on missing premiseExact endpoint-graph modelHigher-rank spectral bicycle channels — Depends on missing premiseHigher-rank spectral bicyclechannelsPositive-defect localization — activePositive-defect localizationUniversal constant-core incompatibility — activeUniversal constant-coreincompatibilityStable bounded affine certificates — activeStable bounded affinecertificatesExternal theorem audit — activeExternal theorem auditSmoothing away the shared vertex — stoppedSmoothing away the sharedvertexInvariant edge-block lemma — stoppedInvariant edge-block lemmaUniversal positive order-defect rank — stoppedUniversal positiveorder-defect rankFirst two branched-double-cover residues — stoppedFirst twobranched-double-coverresiduesHistorical rank-one realizability bridge — BlockedHistorical rank-onerealizability bridgeProve legal two-edge block surgery — OpenProve legal two-edge blocksurgeryLocalize positive-defect residue — OpenLocalize positive-defectresidueExclude every constant-parity square-zero core — OpenExclude everyconstant-parity square-zerocoreProve a stable Gram-or-affine certificate theorem — OpenProve a stableGram-or-affine certificatetheoremAudit both external starting theorems — OpenAudit both external startingtheorems
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.

Active routePositive-defect localization

Use the rank-K residue budget only after it is realized by genuine block-local word features and tied to a strict descent measure.

Route status · Active route
Active routeUniversal constant-core incompatibility

Generalize the exact F₆,₆ exclusions to every square-zero constant-parity order core using matching capacities, Gram infeasibility, kernel separation, and bounded affine certificates.

Route status · Active route
Active routeStable bounded affine certificates

A parallel active route seeks uniform bounded paired-prefix dual witnesses, with exact commuting and rank-13 cores serving as adversarial tests.

Route status · Active route
Active routeExternal theorem audit

The source’s two imported starting theorems must be checked at statement level before any proof claim can use the internal route as a closing argument.

Route status · Active route

Explored alternatives

Other routes

8 recorded
Not yet justifiedHistorical rank-one realizability and block-surgery target

The public baseline did not justify a geometric rank-one channel: the proposed route required either a fully legal word-level surgery or a bounded diagonal-independent affine certificate.

Route status · Not yet justified
Route held in reserveHigher-rank realizable-channel extraction

Higher-rank spectral decomposition remains in the current research map, but word-level descent is paused until an actual chord-neighborhood factor or obstruction can be extracted.

Route status · Route held in reserve
Route held in reservePure edge-removal route

Winding parity controls opposite-cycle blockers, but no mechanism excludes same-cycle vertices from every eligible reduction triangle.

Route status · Route held in reserve
Browse 5 more explored routes
Eliminated routeBranched-double-cover residues

The first two 2-adic stages yield no new obstruction; any continuation would need to encode the global crossing orders rather than only sign and magnitude data.

Route status · Eliminated route
Refuted routeUniversal positive rank of K

Refuted by the exact commuting F₆,₆ order; rank K remains a defect budget rather than a universal obstruction.

Route status · Refuted route
Refuted routeArbitrary principal restriction

Refuted by the reconstructed C₈ word; only clean paired-column deletion, κ-deletion, or complete block surgery remains admissible.

Route status · Refuted route
Refuted routeInvariant module contains an edge block

Refuted by an exact C₆ rank-two module with no whole edge-block indicator.

Route status · Refuted route
Useful but insufficientScalar and low-moment shortcuts

Scalar Arf comparison and the first mixed moments add no information beyond the existing parity and invariant-space equations.

Route status · Useful but insufficient

Route statements and reductions

Statements the next route can inspect and build on

Route statementAffine obstruction and dual certificate

Every geometric endpoint placement lies in an exact row-wise affine hull, and a Frobenius-dual block-constant witness can rule out every placement and diagonal completion in that relaxation.

Source-reported route statement · dependencies incomplete
Route statementConstant parity forces a square-zero core

When both cycle-side endpoint parities are constant, K² = 0, RK = SK = 0, and the joint (E,F)-module decomposes into one-dimensional simultaneous eigenspaces and two-dimensional rank-one defect blocks.

Source-reported route statement · dependencies incomplete
Route statementPositive-defect bicycle purification

For arbitrary K, the parity interface admits a noncanonical decomposition B = B_bic + B_vis in which B_bic factors through bicycle spaces and rank(B_vis) ≤ rank K; this is not a partition into letters or an induced subword.

Source-reported route statement · dependencies incomplete
Route statementThree retained square-zero F₆,₆ cores excluded

The current work’s exact linear Gram ranks and affine certificates exclude every constant-parity sector for the retained rank-15, rank-13, and commuting K=0 F₆,₆ order cores.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks
01
Exclude every constant-parity square-zero core

Prove that no block-respecting square-zero order pair supports both complete interval systems in any of the four constant-parity sectors.

Suggested move: Dualize the linear Gram system symbolically and test a bounded paired-prefix certificate theorem against the rank-13 coupled-row adversary.
Ready to work on
02
Audit both external starting theorems

Verify the exact minimal-counterexample reduction to even–even figure-eights and the sphere/plane hypotheses of the interlacement idempotent-completion criterion for the distinguished word.

Suggested move: Perform statement-level literature checks before any closing claim and bind the exact theorem versions and hypotheses.
Ready to work on
03
Localize positive-defect residue

Realize B_vis by O(rank K) genuine block-local exceptions and prove that deleting, charging, or certifying them strictly decreases a valid finite measure.

Suggested move: Classify actual word features carrying the bounded correction signatures and keep B_bic attached to the original system until a word operation is proved.
Ready to work on
04
Prove a stable Gram-or-affine certificate theorem

Show that every square-zero low-defect order pair either fails the linear Gram system or contains a bounded paired-prefix core with a diagonal-independent affine certificate stable under harmless order perturbations.

Suggested move: Use the commuting, rank-15, and rank-13 F₆,₆ cores as exact adversarial tests and retain every certificate support.
Ready to work on
05
Prove legal two-edge block surgery

Package local complementations and marker cancellations into deletion of two cycle edges while preserving the full distinguished-word, interval, normal-form, coupling, and induction data.

Suggested move: Write the complete before/after word for the smallest pure interface and verify all eight conditions in Section 33.1, including the updates to E, F, Z, K, R, S, A, Π, U, and ∂.
Ready to work on
06
Historical rank-one realizability bridge

At the public baseline, converting a spectral rank-one factor into an actual chord-neighborhood or twin module remained blocked because spectral factors need not be realized by word geometry.

Suggested move: Require an actual word-level realization or a bounded affine certificate; do not treat a spectral summand as a literal chord neighborhood.
Blocked by the current route

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026
Current statusOpen conjecture

The conjecture that every thrackle has at most as many edges as vertices remains open. The best published general linear bound identified is m <= 1.393(n-1).

[3][5]
External progress

What the literature has established

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

  1. PreprintThe theory was extended to nonplanar surfaces while the planar conjecture and 1.393n general record remained open/current in the source's account.[3]
  2. Peer reviewedXu improved the general upper bound to m <= 1.393(n-1).[5]
  3. PreprintFulek and Pach gave a computer-assisted 1.428n bound and an algorithm that, for any epsilon > 0, either proves a (1+epsilon)n bound or exhibits a counterexample.[1]
  4. Peer reviewedLovász, Pach, and Szegedy proved a first linear bound and established the conjecture for x-monotone thrackles.[4][8]
12 cited sources4 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusConway's thrackle conjecture
Equivalent formulationpseudoforest characterization of thrackleable graphs

If Conway's conjecture holds, Woodall's structural results characterize thrackleable graphs as a restricted pseudoforest class.

[6][1]
Weaker or relaxed formgeneralized thrackles

Generalized thrackles require pairs of edges to meet an odd number of times rather than exactly once; every thrackle is generalized, but not conversely.

[1]
Solved special casex-monotone thrackles

Conway's inequality is known when every edge is x-monotone.

[8]
Solved special casespherical thrackles

The conjecture holds for spherical thrackles whose edges are great-circle arcs.

[2]

Formal and computational footholds

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

  • computation · source linked; not reproduced by ProofAtlasbounded graph-planarity search

    Fulek and Pach give a finite algorithm based on testing planarity of derived graphs; it ruled out specified dumbbell graphs and supports quantitative upper bounds, but does not settle the conjecture.

    [1]

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

10 mapped milestonesretained argument map

Browse all 10 mapped stages

  1. stage 1Even–even figure-eight target
  2. stage 2Distinguished-word normal form
  3. stage 3Endpoint and affine obstruction machinery
  4. stage 4Universal positive-rank route refuted
  5. stage 5C₆ zero-defect parity rigidity
  6. stage 6C₈ reconstruction supersedes the broad parity-ear target
  7. stage 7Endpoint-bicycle purification
  8. stage 8Rank-one and higher-rank channel structure
  9. stage 9Three retained square-zero cores excluded
  10. stage 10Legal block-surgery frontier
Even–even figure-eight targetThe retained route conditionally reduces the conjecture to even–even figure-eights and internally narrows those to p,q at least six.

Mapped research milestoneInitial research sequence

Research stage 1
Distinguished-word normal formThe shared vertex is recorded as a distinguished chord and the resulting word equations reduce to an exact two-cycle F₂ normal form.

Mapped research milestoneInitial research sequence

Research stage 2
Endpoint and affine obstruction machineryExact endpoint graphs and affine row hulls turn interval placement into local linear obstruction and certificate problems.

Mapped research milestoneInitial research sequence

Research stage 3
Universal positive-rank route refutedAn exact commuting F₆,₆ order has K = 0, so rank K cannot be a universal obstruction.

Mapped research milestoneInitial research sequence

Research stage 4
C₆ zero-defect parity rigidityThe parity-defect identity plus the eight-word classification forces constant endpoint parity on every defect-zero C₆ side.

Mapped research milestoneInitial research sequence

Research stage 5
C₈ reconstruction supersedes the broad parity-ear targetThe exact C₈ witness shows the isolated ear is charged and refutes arbitrary six-block principal restriction.

Mapped research milestoneInitial research sequence

Research stage 6
Endpoint-bicycle purificationThe exact defect hierarchy identifies endpoint bicycles as the zero-defect ghost space and bounds the positive-defect visible residue by rank K.

Mapped research milestoneInitial research sequence

Research stage 7
Rank-one and higher-rank channel structurePure and charged rank-one interfaces are classified, and zero-defect higher rank splits algebraically into independent opposite-eigenvalue channels.

Mapped research milestoneInitial research sequence

Research stage 8
Three retained square-zero cores excludedExact Gram ranks and affine certificates exclude every constant-parity sector of the retained rank-15, rank-13, and commuting F₆,₆ cores.

Mapped research milestoneInitial research sequence

Research stage 9
Legal block-surgery frontierThe first missing internal theorem is now a realizable rank-one channel plus legal two-edge block surgery, or a bounded affine certificate.

Mapped research milestoneInitial research sequence

Research stage 10

Detailed research inventory

Claims, milestones, and routes in the current map

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

20 standing statements1 proposed statements10 mathematical milestones6 open questions7 conditional results3 completed special cases
Statements by mathematical role21 selected mapped statements
  • theorem candidate1 of 211
  • reduction3 of 213
  • lemma10 of 2110
  • negative result2 of 212
  • equivalence3 of 213
  • counterexample1 of 211
  • computational claim1 of 211
Selected mathematical clusters7 mathematical clusters
Problem and figure-eight reductionThe open conjecture, conditional starting reduction, local topology, and smallest side exclusion.8 displayed rows · 2 routes included
  • retained route statementConway’s thrackle conjecture
  • retained route statementEven–even figure-eight reductionconditional
  • retained route statementNonalternation at the shared vertexconditional
  • retained route statementFour-cycle side excludedspecial case
  • ChallengeThe source imports the minimal-counterexample/doubling reduction without reproducing or auditing its exact hypotheses.unsupported step · open
  • Research targetAudit both external starting theoremsopen
  • Route held in reservePure edge-removal routeWinding parity controls opposite-cycle blockers, but no mechanism excludes same-cycle vertices from every eligible reduction triangle.
  • Active routeExternal theorem auditThe source’s two imported starting theorems must be checked at statement level before any proof claim can use the internal route as a closing argument.
Distinguished word and exact algebraThe marked shared vertex, full interlacement system, canonical projections, and exact F₂ normal form.5 displayed rows · 1 route included
  • retained route statementDistinguished-word equivalenceconditional
  • retained route statementExact F₂ normal formintermediate
  • DerivationThe grouped distinguished word fixes the block interlacement matrix, and expanding a symmetric idempotent completion yields the canonical projections, Z, K, and the two-cycle Gram equations.active reported
  • ChallengeThe exact Rosenstiehl/de Fraysseix criterion and its sphere/plane hypotheses for this distinguished one-component word remain to be checked.unsupported step · open
  • Active routeExternal theorem auditThe source’s two imported starting theorems must be checked at statement level before any proof claim can use the internal route as a closing argument.
Endpoint geometry and affine certificatesExact cut-space interval structure and local dual infeasibility machinery.5 displayed rows · 1 route included
  • retained route statementExact endpoint-graph modelintermediate
  • retained route statementAffine obstruction and dual certificateintermediate
  • DerivationThe exact affine hull of interval rows turns the intertwining equation into a linear image-membership condition; Frobenius duality converts its failure into a local block-constant certificate.active reported
  • Research targetProve a stable Gram-or-affine certificate theoremopen
  • Active routeStable bounded affine certificatesA parallel active route seeks uniform bounded paired-prefix dual witnesses, with exact commuting and rank-13 cores serving as adversarial tests.
Parity and defect hierarchyFirst-order parity, exact higher defects, decoupling, and low-rank purification.8 displayed rows · 1 route included
  • retained route statementParity-defect identityintermediate
  • retained route statementC₆ parity rigidityspecial case
  • retained route statementExact C₈ reconstruction and charged earcomputational
  • retained route statementExact defect hierarchyintermediate
  • retained route statementParity decoupling and low-rank repairintermediate
  • retained route statementPositive-defect bicycle purificationintermediate
  • Research targetLocalize positive-defect residueopen
  • Active routePositive-defect localizationUse the rank-K residue budget only after it is realized by genuine block-local word features and tied to a strict descent measure.
Endpoint bicycles and channel surgeryZero-defect bicycle spaces, rank-one structure, higher-rank algebraic channels, κ-deletion, and the missing word-compatible block surgery.10 displayed rows · 2 routes included
  • retained route statementEndpoint bicycles are the zero-defect ghost spaceconditional
  • retained route statementRank-one parity-interface classificationconditional
  • retained route statementHigher-rank spectral bicycle channelsconditional
  • retained route statementIdempotent local-complement deletionintermediate
  • Research targetHistorical rank-one realizability bridgeblocked
  • Research targetProve legal two-edge block surgeryopen
  • Useful failureTreating algebraic channels as literal chord modulesreported failure
  • Useful failureUsing one κ-deletion as a graph-edge reductionreported failure
  • Not yet justifiedHistorical rank-one realizability and block-surgery targetThe public baseline did not justify a geometric rank-one channel: the proposed route required either a fully legal word-level surgery or a bounded diagonal-independent affine certificate.
  • Route held in reserveHigher-rank realizable-channel extractionHigher-rank spectral decomposition remains in the current research map, but word-level descent is paused until an actual chord-neighborhood factor or obstruction can be extracted.
Constant-parity square-zero coresSquare-zero reduction, exact F₆,₆ adversarial cores, Gram feasibility, and affine certificates.9 displayed rows · 2 routes included
  • retained route statementConstant parity forces a square-zero coreconditional
  • retained route statementCommuting F₆,₆ order with K = 0computational
  • retained route statementThree retained square-zero F₆,₆ cores excludedspecial case
  • ComputationExact Appendix A check of the block-respecting commuting F₆,₆ order and its affine dual certificates.The source reports K = 0, four simultaneous eigenspaces of dimension eight, a seven-entry certificate for Word 1, and certificate existence for all eight admissible C₆ words. · reported unreproduced
  • ComputationAppendix B exact rank and certificate checks for rank-15 and rank-13 square-zero F₆,₆ cores.The source reports K² = 0 with ranks 15 and 13, a one-row kernel-separation witness for rank 15, and a six-entry coupled affine certificate for rank 13, with certificates for all eight C₆ words on the retained side. · reported unreproduced
  • ComputationConstant-parity linear Gram feasibility ranks for the rank-15, rank-13, and commuting F₆,₆ order cores in all four parity sectors.The reported rank/augmented-rank table excludes all sectors for the rank-15 and rank-13 cores and three sectors for the commuting core; the surviving commuting even-even sector is excluded by affine certificates. · reported unreproduced
  • Research targetExclude every constant-parity square-zero coreopen
  • Active routeUniversal constant-core incompatibilityGeneralize the exact F₆,₆ exclusions to every square-zero constant-parity order core using matching capacities, Gram infeasibility, kernel separation, and bounded affine certificates.
  • Refuted routeUniversal positive rank of KRefuted by the exact commuting F₆,₆ order; rank K remains a defect budget rather than a universal obstruction.
Failed, paused, and narrowed routesExact counterexamples and scope warnings prevent renewed work on smoothed words, invariant blocks, rank-only closure, principal deletion, scalar shortcuts, and unrealized algebraic channels.15 displayed rows · 5 routes included
  • Useful failureSmoothing away the shared vertexreported failure
  • Useful failureInvariant edge-block lemmareported failure
  • Useful failureUniversal positive order-defect rankreported failure
  • Useful failureFirst two branched-double-cover residuesreported failure
  • Useful failureArbitrary principal or six-block deletionreported failure
  • Useful failureTreating algebraic channels as literal chord modulesreported failure
  • Useful failureUsing one κ-deletion as a graph-edge reductionreported failure
  • Useful failureTreating bicycle purification as an induced subwordreported failure
  • Useful failureScalar Arf invariants and low mixed momentsreported failure
  • ComputationSource-reported fourth-audit rerun of both embedded Python verification programs.The current work reports that both scripts ran unchanged under Python 3.13.5 and NumPy 2.3.5 and that all assertions and terminal summaries matched expectations; this page did not independently rerun them. · reported unreproduced
  • Eliminated routeBranched-double-cover residuesThe first two 2-adic stages yield no new obstruction; any continuation would need to encode the global crossing orders rather than only sign and magnitude data.
  • Refuted routeUniversal positive rank of KRefuted by the exact commuting F₆,₆ order; rank K remains a defect budget rather than a universal obstruction.
  • Refuted routeArbitrary principal restrictionRefuted by the reconstructed C₈ word; only clean paired-column deletion, κ-deletion, or complete block surgery remains admissible.
  • Refuted routeInvariant module contains an edge blockRefuted by an exact C₆ rank-two module with no whole edge-block indicator.
  • Useful but insufficientScalar and low-moment shortcutsScalar Arf comparison and the first mixed moments add no information beyond the existing parity and invariant-space equations.
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 bridgeProve that no block-respecting square-zero order pair supports both complete interval systems in any of the four constant-parity sectors.

The current research map records this as an open mathematical step.

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.

  • Exclude all square-zero order pairs, not only the three retained F₆,₆ examples, using matching capacities, Gram infeasibility, or stable affine certificates.

Continue the mathematics

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Prepared starting pointExclude every constant-parity square-zero core

Conway’s Thrackle Conjecture · ready to start

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Research contextPrepared context for any AI agent

In a drawing where every pair of distinct edges meets exactly once, the conjecture says the graph cannot have more edges than vertices.

  • 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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Sources and references12 cited works · next context review by Nov 2, 2026

The mathematical context was checked on Aug 2, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    A computational approach to Conway's thrackle conjectureoriginal source · accessed Aug 2, 2026
  2. 2
    Spherical Thracklespreprint · accessed Aug 2, 2026
  3. 3
    Thrackles on nonplanar surfacespreprint · accessed Aug 2, 2026
  4. 4
    On Conway's Thrackle Conjecturepeer reviewed result · accessed Aug 2, 2026
  5. 5
    A New Upper Bound for Conway's Thracklespeer reviewed result · accessed Aug 2, 2026
  6. 6
    Thrackleencyclopedia · accessed Aug 2, 2026
  7. 7
    Formal Conjectures repositoryformalization · accessed Aug 2, 2026
  8. 8
    Conway's conjecture for monotone thracklesauthoritative webpage · accessed Aug 2, 2026
  9. 9
    Five $1,000 Problems (Update 2017)authoritative webpage · accessed Aug 2, 2026
  10. 10
    TOPP: Problem 30 — Thracklesmaintained problem list · accessed Aug 2, 2026
  11. 11
    The Thrackle Problemoriginal source · accessed Aug 2, 2026
  12. 12
    Conway's Thrackle Conjecturemaintained problem list · Epoch AI · accessed Aug 2, 2026

Important qualifications

  • Sources say Conway posed the problem in the late 1960s or about forty years before 1997, but no exact year was verified. The value is therefore null.
  • No authoritative public formalization was located in the scoped Formal Conjectures/Mathlib searches. An empty array is not a claim that none exists.
  • Empty formalization or computation lists mean that none was verified in this scoped search, not that none exists.
  • FrontierMath membership is mutable and must be rechecked after source changelog updates; membership is not proof evidence or an importance score.

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