Describes how distinct surgery presentations of a given 3-manifold are related
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-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". 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-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 pair of 3-balls (the attaching region of the 1-handle) or, more commonly,
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.[1] Excising this 2-handle is equivalent to adding a 1-handle; 3-handles and 4-handles are usually not indicated in the diagram.
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In mathematics, the Kirbycalculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite...
England Kirby High School (disambiguation), several schools in the United States KirbycalculusKirby (cucumber), used for pickling Kirby 23, a sailboat...
topological manifold. He also proved the fundamental result on the Kirbycalculus, a method for describing 3-manifolds and smooth 4-manifolds by surgery...
MR0149457 Gompf, Robert E.; Stipsicz, András I. (1999). 4-manifolds and Kirbycalculus. Graduate Studies in Mathematics. Vol. 20. Providence, RI: American...
presentations of a given 3-manifold might be related. The answer is called the Kirbycalculus. Hyperbolic Dehn surgery Tubular neighborhood Surgery on manifolds,...
change the crossing number of the diagram. In applications such as the Kirbycalculus, in which the desired equivalence class of knot diagrams is not a knot...
smooth manifolds. Casson handle Cobordism theory CW complex Handlebody Kirbycalculus Manifold decomposition S. Smale, "On the structure of manifolds" Amer...
group Writhe Quandle Seifert surface Braids Braid theory Braid group Kirbycalculus Genus (mathematics) Examples Positive Euler characteristic 2-disk Sphere...
ISBN 978-0-8218-2055-1. Gompf, Robert E.; Stipsicz, Andras I. (1999). 4-Manifolds and KirbyCalculus. American Mathematical Society. pp. 55–58, 186–187. ISBN 0-8218-0994-6...
cork. Gompf, Robert E.; Stipsicz, András I. (1999). 4-manifolds and Kirbycalculus. Graduate Studies in Mathematics. Vol. 20. Providence, RI: American...
the above-mentioned applications, Robion Kirby used Cerf Theory as a key step in justifying the Kirbycalculus. A stratification of the complement of an...
C. Evans (2010, 2nd ed., ISBN 978-0-8218-4974-3) 20 4-Manifolds and KirbyCalculus, Robert E. Gompf, András I. Stipsicz (1999, ISBN 978-0-8218-0994-5)...
Mumbai in 2006 for a thesis titled "Knots, mapping class groups and Kirbycalculus", and MSc degree from Maharaja Sayajirao University of Baroda, Vadodara...
and symplectic topology). With András I. Stipsicz: 4-manifolds and Kirbycalculus, AMS 1999 A new construction of symplectic manifolds, Annals of Mathematics...
allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within...
Finite element exterior calculus (FEEC) is a mathematical framework that formulates finite element methods using chain complexes. Its main application...
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