Ambient isotopy

2024-03-24T22:23:52Z

2.100.56.85: Undid revision 1215398669 by 2.100.56.85 (talk)

{{Short description|Concept in toplogy}}
{{multiple image
| total_width = 320
| image1 = Blue Unknot.png
| image2 = Blue Trefoil Knot.png
| footer = In <math>\mathbb{R}^3</math>, the [[unknot]] is not '''ambient-isotopic''' to the [[trefoil knot]] since one cannot be deformed into the other through a continuous path of homeomorphisms of the ambient space. They are ambient-isotopic in <math>\mathbb{R}^4</math>.
}}
In the [[mathematics|mathematical]] subject of [[topology]], an '''ambient isotopy''', also called an ''h-isotopy'', is a kind of [[continuous map|continuous]] distortion of an [[ambient space]], for example a [[manifold]], taking a [[submanifold]] to another submanifold. For example in [[knot theory]], one considers two [[knot (mathematics)|knot]]s the same if one can distort one knot into the other without breaking it. Such a distortion is an example of an ambient isotopy. More precisely, let <math>N</math> and <math>M</math> be manifolds and <math>g</math> and <math>h</math> be [[embedding]]s of <math>N</math> in <math>M</math>. A [[continuous map]]
:<math>F:M \times [0,1] \rightarrow M </math>
is defined to be an ambient isotopy taking <math>g</math> to <math>h</math> if <math>F_0</math> is the [[identity function|identity map]], each map <math>F_t</math> is a [[homeomorphism]] from <math>M</math> to itself, and <math>F_1 \circ g = h</math>. This implies that the [[orientation (geometry)|orientation]] must be preserved by ambient isotopies. For example, two knots that are [[mirror image]]s of each other are, in general, not equivalent.

==See also==
*[[Homotopy#Isotopy|Isotopy]]
*[[Regular homotopy]]
*[[Regular isotopy]]

== References ==
*M. A. Armstrong, ''Basic Topology'', [[Springer-Verlag]], 1983
*Sasho Kalajdzievski, ''An Illustrated Introduction to Topology and Homotopy'', CRC Press, 2010, Chapter 10: Isotopy and Homotopy
[[Category:Topology]]
[[Category:Maps of manifolds]]

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Ambient isotopy