Kirby calculus
Encyclopedia
In mathematics
, the Kirby calculus in geometric topology
, named after Robion Kirby
, is a method for modifying framed links in the 3-sphere
using a finite set of moves, the Kirby moves. Using four-dimensional Cerf theory
, he proved that if M and N are 3-manifold
s, 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". 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"; 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). 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 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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Kirby calculus in geometric topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
, named after Robion Kirby
Robion Kirby
Robion Cromwell Kirby is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology...
, is a method for modifying framed links in the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
using a finite set of moves, the Kirby moves. Using four-dimensional Cerf theory
Cerf theory
In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functionsf:M \to \mathbb R...
, he proved that if M and N are 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s, resulting from Dehn surgery
Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...
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
Lickorish–Wallace theorem
In mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ±1 surgery coefficients. Furthermore, each component of the link can be assumed to...
any closed
Closed
Closed may refer to:Math* Closure * Closed manifold* Closed orbits* Closed set* Closed differential form* Closed map, a function that is closed.Other* Cloister, a closed walkway* Closed-circuit television...
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"; 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). 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
Slam-dunk
In mathematics, particularly low-dimensional topology, the slam-dunk is a particular modification of a given surgery diagram in the 3-sphere for a 3-manifold. The name, but not the move, is due to Tim Cochran. Let K be a component of the link in the diagram and J be a component that circles K as...
.
An extended set of diagrams and moves are used for describing 4-manifold
4-manifold
In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...
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 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.
Handle decomposition
- A closed, smooth 4-manifold 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-ballsBall (mathematics)In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
. - A 2-handle is attached along a solid torusSolid torusIn mathematics, a solid torus is a topological space homeomorphic to S^1 \times D^2, i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary...
; since this solid torus is embedded in a 3-manifold3-manifoldIn mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
, there is a relation between handle decompositions on 4-manifolds, and knot theoryKnot theoryIn topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
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.