{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.betti-number","code":"thing-q429593","url":"https://ver.cy/models/thing/q429593/","name":"Betti number","alternateNames":[],"kind":"thing","status":"published","version":"0.2.0-wave.3","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QTY"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A topological invariant that counts the number of independent holes of each dimension in a space, the k-th Betti number being the rank of the k-th homology group, so that b0 counts connected components, b1 counts one-dimensional holes or loops and b2 counts enclosed voids, with the alternating sum of Betti numbers giving the Euler characteristic; Betti numbers are computed for surfaces, manifolds, simplicial complexes and data in topological data analysis.","purpose":"Let an agent explain Betti numbers and their meaning, relay definitions, examples and computation from algebraic topology references, describe applications including topological data analysis, and distinguish Betti numbers from the Euler characteristic and homology groups themselves.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Counts holes by dimension","Homological rank","Topological invariance","Data analysis use"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What a Betti number is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is the k-th Betti number, and how does it count components, loops and voids?","kind":"definition"},{"text":"Is the question about Betti numbers, homology groups or the Euler 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Betti numbers depend on the coefficient field, as for the projective plane?","kind":"provenance"},{"text":"Which entry fits homology with coefficients?","kind":"action"}]}]}]},{"id":"apply","name":"Apply","description":"Applications.","layers":[{"id":"tda","name":"TDA","description":"Topological data analysis.","findings":[{"id":"tda-finding","name":"TDA","description":"TDA.","questions":[{"text":"How do persistent Betti numbers and barcodes describe the shape of data?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]},{"id":"science","name":"Science","description":"Science and algebra.","findings":[{"id":"science-finding","name":"Science","description":"Science.","questions":[{"text":"How are Betti numbers used in physics, materials science and commutative algebra?","kind":"provenance"},{"text":"Which entry fits persistent homology?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History and theory.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did Betti and Poincare introduce these numbers, and how did Noether reframe them as ranks of groups?","kind":"provenance"},{"text":"Which entry fits the history of topology?","kind":"action"}]}]},{"id":"theory","name":"Theory","description":"Theory.","findings":[{"id":"theory-finding","name":"Theory","description":"Theory.","questions":[{"text":"How do Betti numbers relate to cohomology, Poincare duality and Morse theory?","kind":"provenance"},{"text":"Which entry fits algebraic topology?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Explain Betti numbers and compute them for spaces given by a user.","State the coefficient field used and why results can depend on it.","Explain persistent Betti numbers in topological data analysis and their limits."],"mustNot":["Confuse Betti numbers with homology groups or the Euler characteristic.","Ignore torsion or coefficient effects when they matter.","Overinterpret persistent Betti numbers as proof of structure in data.","Claim new theorems without proof."],"requiresHuman":["Relying on topological data analysis for medical or financial decisions."]},"ethics":{"considerations":["Topological methods are increasingly used on real data, where overinterpretation can mislead.","Accurate teaching matters for students."],"affectedParties":["Students","Researchers","People affected by data-driven decisions"]},"owners":{"steward":"Nobody owns the concept; the mathematical community maintains it.","roles":[],"masterSystems":[]},"relations":[{"target":"Q28920052","type":"parent","note":"registry parent class"},{"target":"non-negative integer","type":"related","note":"in registry terms"},{"target":"homology group","type":"related","note":"in each dimension"},{"target":"Euler characteristic","type":"related","note":"by alternating sum"},{"target":"topological data analysis","type":"related","note":"through persistence"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.betti-number","Wikidata: Q429593 (https://www.wikidata.org/wiki/Q429593)"]},"properties":{"applicability":"not-applicable","items":["sphere Betti numbers: 1, 0, 1 sequence","torus Betti numbers: 1, 2, 1 sequence","named after: Enrico Betti person - by Poincare"]},"recognition":{"applicability":"optional","items":["Ranks of homology groups","b0 components, b1 loops, b2 voids, higher Betti numbers","The Euler characteristic is their alternating sum; homology groups carry torsion too","Not a visible object; a sequence of non-negative integers."]},"capabilities":{"applicability":"required","items":["explain the definition","relay examples and computation","describe applications","distinguish from related invariants"]},"hazards":{"applicability":"required","items":["Confusing Betti numbers with homology groups or Euler characteristic","Ignoring dependence on coefficient field","Misreading persistent Betti numbers"]},"interfaces":{"applicability":"required","items":["No regulation; standard mathematical definitions"]},"context":{"applicability":"required","items":["Not applicable; a mathematical invariant.","Betti numbers of surfaces and manifolds","Betti numbers of simplicial and CW complexes","Betti numbers over different coefficient fields","persistent Betti numbers in topological data analysis","graded Betti numbers in commutative algebra"]}},"sources":[{"title":"Wikidata item Q429593: Betti number","url":"https://www.wikidata.org/wiki/Q429593","note":"identity and sense of the item"},{"title":"Wikipedia: Betti number","url":"https://en.wikipedia.org/wiki/Betti_number","note":"general description of the item"}],"openQuestions":["Should persistent homology be a separate primary entry?","How should topology references be linked?","How should graded Betti numbers in algebra be recorded?"],"resources":{"spec":"/models/things/publications/thing-q429593/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q429593/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T03:52:47Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-06T19:52:00+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}