Kirby calculus
E911226
Kirby calculus is a set of moves and techniques in low-dimensional topology used to manipulate framed links in three-manifolds and study and classify four-manifolds.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theory
ⓘ
technique in low-dimensional topology ⓘ |
| appliesTo |
3-manifolds
ⓘ
4-manifolds ⓘ |
| basedOn | handle decompositions of manifolds ⓘ |
| characterizedBy | finite set of local moves on framed links ⓘ |
| concerns |
framing integers on link components
ⓘ
linking matrices ⓘ |
| coreConcept |
Kirby moves
NERFINISHED
ⓘ
framed links in S^3 ⓘ handlebody decompositions ⓘ |
| field | low-dimensional topology ⓘ |
| formalism | link diagrams with framings ⓘ |
| hasMove |
blow down
ⓘ
blow up ⓘ cancellation of handle pairs ⓘ destabilization ⓘ handle slide ⓘ introduction of cancelling handle pairs ⓘ stabilization ⓘ |
| hasTheorem | Kirby theorem NERFINISHED ⓘ |
| helpsCompute | intersection form of a 4-manifold from a link diagram ⓘ |
| helpsConstruct | handle decompositions of 4-manifolds from framed links ⓘ |
| implies | two framed links related by Kirby moves yield homeomorphic 3-manifolds ⓘ |
| introducedBy | Robion Kirby NERFINISHED ⓘ |
| Kirby theorem | equivalence of 3-manifolds via surgery corresponds to Kirby moves on framed links ⓘ |
| mathematicsSubjectClassification |
57M25
ⓘ
57R65 ⓘ |
| namedAfter | Robion Kirby NERFINISHED ⓘ |
| relatedTo |
Fenn–Rourke moves
NERFINISHED
ⓘ
Morse theory NERFINISHED ⓘ Rolfsen moves NERFINISHED ⓘ handle theory ⓘ surgery on links ⓘ |
| relates | framed links in S^3 and 3-manifolds obtained by surgery ⓘ |
| toolFor |
proving equivalence of 3-manifolds obtained by surgery
ⓘ
translating handle decompositions into link diagrams ⓘ |
| typicalAmbientSpace | 3-sphere S^3 NERFINISHED ⓘ |
| usedBy | geometric topologists ⓘ |
| usedFor |
classifying four-manifolds
ⓘ
manipulating framed links in three-manifolds ⓘ studying four-manifolds ⓘ |
| usedIn |
Dehn surgery description of 3-manifolds
ⓘ
classification of simply connected smooth 4-manifolds ⓘ construction of exotic 4-manifolds ⓘ study of intersection forms of 4-manifolds ⓘ study of smooth structures on 4-manifolds ⓘ surgery theory ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.