Dehn twist
E265413
A Dehn twist is a fundamental type of self-homeomorphism of a surface obtained by cutting along a simple closed curve, twisting one side by 360 degrees, and gluing it back, playing a central role in low-dimensional topology and the study of mapping class groups.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Dehn twist canonical | 3 |
| fractional Dehn twist | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2416869 — 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 twist Context triple: [Max Dehn, notableWork, Dehn twist]
-
A.
Conway skein triple (L₊, L₋, L₀)
The Conway skein triple (L₊, L₋, L₀) is a standard configuration of three related link diagrams used in knot theory to express how a link invariant, such as the Conway polynomial, changes under local crossing modifications.
-
B.
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.
-
C.
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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D.
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.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- 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 twist Target entity description: A Dehn twist is a fundamental type of self-homeomorphism of a surface obtained by cutting along a simple closed curve, twisting one side by 360 degrees, and gluing it back, playing a central role in low-dimensional topology and the study of mapping class groups.
-
A.
Conway skein triple (L₊, L₋, L₀)
The Conway skein triple (L₊, L₋, L₀) is a standard configuration of three related link diagrams used in knot theory to express how a link invariant, such as the Conway polynomial, changes under local crossing modifications.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
homeomorphism
ⓘ
mapping class ⓘ self-homeomorphism of a surface ⓘ surface diffeomorphism ⓘ |
| actsOn |
oriented surface
ⓘ
simple closed curve on a surface ⓘ |
| algebraicEffect |
acts as a transvection on homology for nonseparating curves
ⓘ
acts on first homology of the surface ⓘ acts on fundamental group of the surface ⓘ acts trivially on homology for separating curves ⓘ |
| appearsIn |
Dehn–Lickorish theorem
ⓘ
Lickorish ⓘ
surface form:
Lickorish’s generating set for mapping class groups
|
| construction |
cut along a simple closed curve
ⓘ
glue the surface back together ⓘ twist one side by 360 degrees ⓘ |
| definedOn |
embedded annulus around the curve
ⓘ
simple closed curve that is two-sided ⓘ |
| direction |
left-handed Dehn twist
ⓘ
right-handed Dehn twist ⓘ |
| field |
Teichmüller theory
ⓘ
geometric group theory ⓘ geometric topology ⓘ low-dimensional topology ⓘ |
| generalization |
Dehn twist
self-linksurface differs
ⓘ
surface form:
fractional Dehn twist
multitwist ⓘ |
| generates |
mapping class group of a closed orientable surface
ⓘ
mapping class group of a surface with boundary ⓘ |
| inverse | inverse Dehn twist ⓘ |
| inverseProperty | inverse is the twist in the opposite direction ⓘ |
| namedAfter | Max Dehn ⓘ |
| playsRoleIn |
Thurston’s classification of surface diffeomorphisms
ⓘ
surface form:
Nielsen–Thurston classification
classification of surface homeomorphisms ⓘ mapping class group of a surface ⓘ presentation of mapping class groups ⓘ |
| property |
invertible
ⓘ
is identity outside an annular neighborhood of the curve ⓘ orientation-preserving ⓘ supported in an annular neighborhood of the curve ⓘ |
| satisfies |
braid relations with twists about intersecting curves
ⓘ
chain relation on a chain of curves ⓘ commutation relations for disjoint curves ⓘ lantern relation on a sphere with four boundary components ⓘ |
| topologicalType | isotopy class of a homeomorphism ⓘ |
| usedIn |
Lefschetz fibration
ⓘ
surface form:
Picard–Lefschetz theory
cluster algebra combinatorics on surfaces ⓘ construction of Lefschetz fibrations ⓘ construction of pseudo-Anosov homeomorphisms ⓘ monodromy factorizations ⓘ study of 3-manifolds via Heegaard splittings ⓘ surgery descriptions of 3-manifolds ⓘ symplectic topology ⓘ |
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 twist Description of subject: A Dehn twist is a fundamental type of self-homeomorphism of a surface obtained by cutting along a simple closed curve, twisting one side by 360 degrees, and gluing it back, playing a central role in low-dimensional topology and the study of mapping class groups.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
fractional Dehn twist