Dehn twist

E265413

homeomorphism mapping class self-homeomorphism of a surface surface diffeomorphism

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.

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All labels observed (2)

Label	Occurrences
Dehn twist canonical	3
fractional Dehn twist	1

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 ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Max Dehn → notableWork → Dehn twist ⓘ

Max Dehn → notableConcept → Dehn twist ⓘ

Max Dehn → hasEponym → Dehn twist ⓘ

Dehn twist → generalization → Dehn twist self-linksurface differs ⓘ

this entity surface form: fractional Dehn twist