Dehn surgery
E265411
Dehn surgery is a fundamental operation in 3-manifold topology that modifies a 3-dimensional manifold by cutting out a solid torus and gluing it back in a different way, playing a central role in the classification and study of 3-manifolds.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Dehn filling | 4 |
| Dehn surgery canonical | 4 |
| Lickorish–Wallace theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2416867 — 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: Dehn surgery Context triple: [Max Dehn, notableWork, Dehn surgery]
-
A.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
-
B.
Jones polynomial
The Jones polynomial is a powerful knot invariant in topology that assigns to each knot or link a Laurent polynomial, enabling the distinction of many knots that are indistinguishable by classical invariants.
-
C.
Conway sphere
The Conway sphere is a mathematical construct in knot theory used to decompose knots and links into simpler tangles, named after mathematician John Horton Conway.
-
D.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
E.
Dowker–Thistlethwaite notation
Dowker–Thistlethwaite notation is a numerical encoding system used in knot theory to uniquely represent knot diagrams and facilitate their classification and study.
- 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: Dehn surgery Target entity description: Dehn surgery is a fundamental operation in 3-manifold topology that modifies a 3-dimensional manifold by cutting out a solid torus and gluing it back in a different way, playing a central role in the classification and study of 3-manifolds.
-
A.
geometrization conjecture
The geometrization conjecture is a fundamental statement in 3-dimensional topology that classifies all closed 3-manifolds into pieces each admitting one of eight canonical geometric structures, a result proven by Grigori Perelman.
-
B.
Jones polynomial
The Jones polynomial is a powerful knot invariant in topology that assigns to each knot or link a Laurent polynomial, enabling the distinction of many knots that are indistinguishable by classical invariants.
-
C.
Conway sphere
The Conway sphere is a mathematical construct in knot theory used to decompose knots and links into simpler tangles, named after mathematician John Horton Conway.
-
D.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
E.
Dowker–Thistlethwaite notation
Dowker–Thistlethwaite notation is a numerical encoding system used in knot theory to uniquely represent knot diagrams and facilitate their classification and study.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
construction in 3-manifold topology
ⓘ
topological operation ⓘ |
| actsOn | 3-manifold ⓘ |
| alsoKnownAs |
Dehn surgery
ⓘ
surface form:
Dehn filling
|
| canProduce |
Seifert fibered spaces
ⓘ
hyperbolic 3-manifolds ⓘ lens spaces ⓘ |
| centralRoleIn |
geometrization conjecture
ⓘ
surface form:
Geometrization program
classification of closed orientable 3-manifolds ⓘ study of hyperbolic 3-manifolds ⓘ |
| coreIdea | remove a solid torus and reglue it by a homeomorphism of its boundary ⓘ |
| domain | 3-dimensional manifolds ⓘ |
| effect |
changes the topology of the manifold
ⓘ
may change the fundamental group ⓘ may change the geometric structure ⓘ |
| field |
3-manifold topology
ⓘ
geometric topology ⓘ |
| generalizes | lens space construction from surgery on the unknot ⓘ |
| historicalPeriod | early 20th century ⓘ |
| implies | every closed orientable 3-manifold can be obtained by surgery on a link in S^3 ⓘ |
| input |
3-manifold with torus boundary
ⓘ
knot in the 3-sphere ⓘ link in a 3-manifold ⓘ |
| namedAfter | Max Dehn ⓘ |
| output | 3-manifold ⓘ |
| parameterizedBy |
isotopy class of simple closed curve on the boundary torus
ⓘ
rational slope ⓘ |
| relatedConcept |
Dehn filling of a cusped hyperbolic 3-manifold
ⓘ
Kirby calculus ⓘ framing of a knot ⓘ surgery coefficient ⓘ |
| relatedTheorem |
Dehn surgery
self-linksurface differs
ⓘ
surface form:
Lickorish–Wallace theorem
Thurston hyperbolization theorem ⓘ
surface form:
Thurston hyperbolic Dehn surgery theorem
|
| requires | choice of homeomorphism of the boundary torus ⓘ |
| specialCase | knot surgery in S^3 ⓘ |
| specialCaseOf | surgery on a manifold ⓘ |
| step |
choose an embedded solid torus in a 3-manifold
ⓘ
glue back a solid torus via a boundary homeomorphism ⓘ remove the interior of the solid torus ⓘ |
| studiedIn | low-dimensional topology literature ⓘ |
| typicalAmbientSpace |
3-sphere S^3
ⓘ
oriented closed 3-manifold ⓘ |
| usedIn |
construction of counterexamples in 3-manifold topology
ⓘ
study of Dehn fillings of cusped hyperbolic 3-manifolds ⓘ study of knot complements ⓘ |
| usesConcept |
slope on a torus
ⓘ
solid torus ⓘ torus boundary ⓘ |
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: Dehn surgery Description of subject: Dehn surgery is a fundamental operation in 3-manifold topology that modifies a 3-dimensional manifold by cutting out a solid torus and gluing it back in a different way, playing a central role in the classification and study of 3-manifolds.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Dehn filling
this entity surface form:
Dehn filling
subject surface form:
Max Dehn
subject surface form:
Max Dehn
this entity surface form:
Dehn filling
this entity surface form:
Dehn filling
this entity surface form:
Lickorish–Wallace theorem