Kirby calculus

Using four-dimensional Cerf theory, he proved that if M and N are 3-manifolds, resulting from Dehn surgery on framed links L and J respectively, then they are homeomorphic if and only if L and J are related by a sequence of Kirby moves.According to the Lickorish–Wallace theorem any closed orientable 3-manifold is obtained by such surgery on some link in the 3-sphere.Some ambiguity exists in the literature on the precise use of the term "Kirby moves".Kirby's original formulation involved two kinds of move, the "blow-up" and the "handle slide"; Roger Fenn and Colin Rourke exhibited an equivalent construction in terms of a single move, the Fenn–Rourke move, that appears in many expositions and extensions of the Kirby calculus.Dale Rolfsen's book, Knots and Links, from which many topologists have learned the Kirby calculus, describes a set of two moves: 1) delete or add a component with surgery coefficient infinity 2) twist along an unknotted component and modify surgery coefficients appropriately (this is called the Rolfsen twist).
mathematicsgeometric topologyRobion Kirbyframed links3-sphereCerf theory3-manifoldsDehn surgeryhomeomorphicLickorish–Wallace theoremclosedorientableColin RourkeDale Rolfsenslam-dunk4-manifoldshandle decompositionattaching mapsolid torus3-manifoldknot theoryExotic R 4 {\displaystyle \mathbb {R} ^{4}} Kirby, RobionInventiones MathematicaeBibcodeRourke, ColinTopologyGompf, RobertGraduate Studies in Mathematics