- Kirby calculus
In
mathematics , the Kirby calculus ingeometric topology is a method for modifyingframed link s in the3-sphere using a finite set of moves, the Kirby moves. It is named forRobion Kirby . Using four dimensionalCerf theory , he proved that if M and N are3-manifold s, resulting fromDehn surgery on framed links L and J respectively, then they arehomeomorphic if and only if L and J are related by a sequence of Kirby moves. According to theLickorish-Wallace theorem anyclosed 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". Different presentations of "Kirby calculus" have a different set of moves and these are sometimes called Kirby moves. Kirby's original formulation involved two kinds of move, the "blow-up" and the "handle slide"; Fenn and 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. 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 theRolfsen twist ). This allows an extension of the Kirby calculus to rational surgeries.There are also various tricks to modify surgery diagrams. One such useful move is the
slam-dunk .An extended set of diagrams and moves are used for describing
4-manifold s.A framed link in the 3-sphere encodes instructions for attaching 2-handles to the 4-ball.(The 3-dimensional boundary of this manifold is the 3-manifold interpretation of the link diagram mentioned above.)1-handles are denoted by either (a) a pair of 3-balls (the attaching region of the 1-handle) or, more commonly,(b) unknotted circles with dots.The dot indicates that a neighborhood of a standard 2-disk with boundary the dotted circle is tobe excised from the interior of the 4-ball. Excising this 2-handle is equivalent to adding a 1-handle.3-handles and 4-handles are usually not indicated in the diagram.Handle decomposition
* A closed, smooth 4-manifold M is usually described by a
handle decomposition .
* A 0-handle is just a ball, and theattaching map is disjoint union.
* A 1-handle is attached along two disjoint 3-balls.
* A 2-handle is attached along asolid torus ; since this solid torus is embedded in a3-manifold , there is a relation between handle decompositions on 4-manifolds, andknot theory in 3-manifolds.
* A pair of handles with index differing by 1, whose cores link each other in a sufficiently simple way can be cancelled without changing the underlying manifold. Similarly, such a cancelling pair can be created.Two different smooth handlebody decompositions of a smooth 4-manifold are related by a finite sequence of isotopies of the attaching maps, and the creation/cancellation of handle pairs.
ee also
References
* Rob Kirby, "A Calculus for Framed Links in S3". Inventiones Mathematicae, vol. 45 (1978), pp. 35-56.
* R. P. Fenn and C. P. Rourke, "On Kirby's calculus of links". Topology, vol. 18 (1979), pp. 1–15
* Robert Gompf and Andras Stipsicz, "4-Manifolds and Kirby Calculus", (1999) (Volume 20 in "Graduate Studies in Mathematics"), American Mathematical Society, Providence, RI ISBN 0-8218-0994-6
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