van Kampen diagram
E911230
A van Kampen diagram is a planar, combinatorial 2-complex used in combinatorial group theory to visually represent relations in a group presentation and to prove that a word equals the identity.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Dehn diagram | 1 |
| van Kampen diagram canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11215183 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: van Kampen diagram Context triple: [Dehn complex, relatedTo, van Kampen diagram]
-
A.
Reidemeister moves
Reidemeister moves are the three local diagrammatic transformations in knot theory that characterize when two knot or link diagrams represent the same topological knot.
-
B.
Dehn algorithm
The Dehn algorithm is a decision procedure in combinatorial group theory that solves the word problem for certain groups by systematically reducing words using defining relations.
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C.
Dehn complex
The Dehn complex is a topological construction introduced by Max Dehn in the study of group presentations and decision problems, encoding relations of a group as a 2-dimensional cell complex.
-
D.
Kerr Penrose diagram
The Kerr Penrose diagram is a conformal spacetime diagram depicting the causal structure of a rotating (Kerr) black hole, including its event horizons, ergoregions, and extended regions.
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E.
Coxeter–Dynkin diagrams
Coxeter–Dynkin diagrams are graphical representations that encode the structure of reflection groups and root systems, widely used in the classification of regular polytopes, Lie algebras, and symmetries.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: van Kampen diagram Target entity description: A van Kampen diagram is a planar, combinatorial 2-complex used in combinatorial group theory to visually represent relations in a group presentation and to prove that a word equals the identity.
-
A.
Reidemeister moves
Reidemeister moves are the three local diagrammatic transformations in knot theory that characterize when two knot or link diagrams represent the same topological knot.
-
B.
Dehn algorithm
The Dehn algorithm is a decision procedure in combinatorial group theory that solves the word problem for certain groups by systematically reducing words using defining relations.
-
C.
Dehn complex
The Dehn complex is a topological construction introduced by Max Dehn in the study of group presentations and decision problems, encoding relations of a group as a 2-dimensional cell complex.
-
D.
Kerr Penrose diagram
The Kerr Penrose diagram is a conformal spacetime diagram depicting the causal structure of a rotating (Kerr) black hole, including its event horizons, ergoregions, and extended regions.
-
E.
Coxeter–Dynkin diagrams
Coxeter–Dynkin diagrams are graphical representations that encode the structure of reflection groups and root systems, widely used in the classification of regular polytopes, Lie algebras, and symmetries.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
combinatorial 2-complex
ⓘ
planar diagram ⓘ tool in combinatorial group theory ⓘ topological object ⓘ |
| basedOn |
group presentation
ⓘ
set of generators of a group ⓘ set of relators of a group ⓘ |
| boundaryRepresents |
element of the group
ⓘ
word in the generators ⓘ |
| canBeEmbeddedIn | plane ⓘ |
| canBeViewedAs | map of a 2-disc into a presentation complex ⓘ |
| encodes |
factorization of a word into conjugates of relators
ⓘ
proof that a word is equal to the identity ⓘ |
| field |
combinatorial group theory
ⓘ
geometric group theory ⓘ low-dimensional topology ⓘ |
| hasAlternativeName | disc diagram ⓘ |
| hasPart |
2-cells labeled by relators
ⓘ
basepoint on the boundary ⓘ boundary cycle ⓘ edges ⓘ edges labeled by generators and their inverses ⓘ faces ⓘ vertices ⓘ |
| hasProperty |
combinatorial
ⓘ
finite 2-dimensional CW-complex ⓘ planar ⓘ simply connected ⓘ |
| namedAfter | Egbert van Kampen ⓘ |
| relatedTo |
Cayley 2-complex
ⓘ
linked to:
Dehn complex
Dehn diagram ⓘ
linked to:
van Kampen diagram
Dehn function ⓘ isoperimetric function of a group ⓘ presentation complex ⓘ word problem for groups ⓘ |
| satisfies |
each 2-cell boundary label is a relator or its inverse
ⓘ
edge labels are consistent on adjacent faces ⓘ outer boundary label is the given word ⓘ |
| topologicallyEquivalentTo | disc diagram ⓘ |
| usedFor |
constructing null-homotopies of loops in a 2-complex
ⓘ
proving that a word equals the identity in a group ⓘ representing relations in a group presentation ⓘ studying the word problem in groups ⓘ visualizing relations among generators of a group ⓘ |
| usedIn |
combinatorial curvature arguments
ⓘ
hyperbolic group theory ⓘ small cancellation theory ⓘ |
| usedInProofOf |
equivalence of words in a group presentation
ⓘ
van Kampen theorem in algebraic topology ⓘ
linked to:
van Kampen theorem
|
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: van Kampen diagram Description of subject: A van Kampen diagram is a planar, combinatorial 2-complex used in combinatorial group theory to visually represent relations in a group presentation and to prove that a word equals the identity.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: van Kampen diagram