Kirby calculus
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In mathematics, the Kirby calculus in geometric topology is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves. It is named for Robion Kirby. 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.
An extended set of diagrams and moves are used for describing 4-manifolds. 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 to be 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.
[edit] Handle decomposition
- A closed, smooth 4-manifold M is usually described by a handle decomposition.
- A 0-handle is just a ball, and the attaching map is disjoint union.
- A 1-handle is attached along two disjoint 3-balls.
- A 2-handle is attached along a solid torus; since this solid torus is embedded in a 3-manifold, there is a relation between handle decompositions on 4-manifolds, and knot 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.
[edit] See also
[edit] References
- Rob Kirby, "A Calculus for Framed Links in S3". Inventiones Mathematicae, vol. 45, 1978, pgs. 539-549.
- 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
