Fenchel–Nielsen coordinates
E898483
Fenchel–Nielsen coordinates are a system of parameters that describe hyperbolic structures on surfaces by recording lengths and twist angles along a collection of simple closed geodesics.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Fenchel–Nielsen coordinates canonical | 6 |
How this entity was disambiguated
This entity first appeared as the object of triple T10991623 — 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: Fenchel–Nielsen coordinates Context triple: [Teichmüller theory, usesConcept, Fenchel–Nielsen coordinates]
-
A.
Milnor–Wood inequality
The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
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B.
Hyperbolic Manifolds and Discrete Groups
"Hyperbolic Manifolds and Discrete Groups" is a foundational mathematical monograph that develops the theory of hyperbolic geometry and its deep connections with discrete group actions and low-dimensional topology.
-
C.
Culler–Vogtmann Outer space
Culler–Vogtmann Outer space is a topological space that parametrizes marked metric graphs, serving as an analogue of Teichmüller space for studying the outer automorphism group of a free group.
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D.
Teichmüller curve
A Teichmüller curve is a complex geodesic in the moduli space of Riemann surfaces that arises from flat surface structures and has rich connections to dynamics, geometry, and number theory.
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E.
Dehn surgery
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.
- 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: Fenchel–Nielsen coordinates Target entity description: Fenchel–Nielsen coordinates are a system of parameters that describe hyperbolic structures on surfaces by recording lengths and twist angles along a collection of simple closed geodesics.
-
A.
Milnor–Wood inequality
The Milnor–Wood inequality is a result in differential geometry and topology that bounds the Euler class of flat circle bundles over surfaces, with important implications for foliations and group actions on the circle.
-
B.
Hyperbolic Manifolds and Discrete Groups
"Hyperbolic Manifolds and Discrete Groups" is a foundational mathematical monograph that develops the theory of hyperbolic geometry and its deep connections with discrete group actions and low-dimensional topology.
-
C.
Culler–Vogtmann Outer space
Culler–Vogtmann Outer space is a topological space that parametrizes marked metric graphs, serving as an analogue of Teichmüller space for studying the outer automorphism group of a free group.
-
D.
Teichmüller curve
A Teichmüller curve is a complex geodesic in the moduli space of Riemann surfaces that arises from flat surface structures and has rich connections to dynamics, geometry, and number theory.
-
E.
Dehn surgery
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Teichmüller theory concept
ⓘ
coordinate system ⓘ hyperbolic geometry concept ⓘ parameterization ⓘ |
| appliesTo |
Riemann surfaces of genus g ≥ 2
ⓘ
hyperbolic surfaces ⓘ oriented surfaces of finite type ⓘ |
| assumes |
complete hyperbolic metric of finite area
ⓘ
geodesic boundary or cusps ⓘ |
| basedOn | pants decomposition of a surface ⓘ |
| codomain | Euclidean space ⓘ |
| defines |
length parameters
ⓘ
twist parameters ⓘ |
| dimensionMatches | dimension of Teichmüller space ⓘ |
| domain |
Teichmüller space of a surface
ⓘ
linked to:
Teichmüller space
|
| forSurfaceOfGenus | g with n boundary components ⓘ |
| generalizes | Fenchel’s description of hyperbolic polygons ⓘ |
| hasAlternativeFormulation | complex Fenchel–Nielsen coordinates ⓘ |
| hasComponent |
length coordinate
ⓘ
twist coordinate ⓘ |
| lengthParameterRepresents | hyperbolic length of a simple closed geodesic ⓘ |
| namedAfter |
Jakob Nielsen
ⓘ
Werner Fenchel ⓘ
linked to:
W. Fenchel
|
| numberOfLengthParameters | 3g - 3 + n ⓘ |
| numberOfTwistParameters | 3g - 3 + n ⓘ |
| property |
coordinates are real-valued
ⓘ
depend on choice of pants decomposition ⓘ give a global real-analytic homeomorphism from Teichmüller space to Euclidean space ⓘ |
| relatedTo |
Weil–Petersson symplectic form
ⓘ
linked to:
Weil–Petersson metric
mapping class group ⓘ moduli space of Riemann surfaces ⓘ |
| totalNumberOfParameters | 6g - 6 + 2n ⓘ |
| twistParameterMeasuredAlong | simple closed geodesic ⓘ |
| twistParameterRepresents | relative twist of pairs of pants along a geodesic ⓘ |
| usedFor |
describing hyperbolic structures on surfaces
ⓘ
describing metrics of constant negative curvature ⓘ parameterizing Teichmüller space ⓘ studying moduli of Riemann surfaces ⓘ |
| usedIn |
complex analysis on Riemann surfaces
ⓘ
geometric group theory ⓘ low-dimensional topology ⓘ study of Kleinian groups ⓘ |
| usesConcept |
Dehn twist
ⓘ
geodesic length ⓘ hyperbolic metric ⓘ simple closed geodesics ⓘ twist parameter ⓘ |
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: Fenchel–Nielsen coordinates Description of subject: Fenchel–Nielsen coordinates are a system of parameters that describe hyperbolic structures on surfaces by recording lengths and twist angles along a collection of simple closed geodesics.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.