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| In the [[mathematics|mathematical]] subject of [[topology]], an '''ambient isotopy''', also called an ''h-isotopy'', is a kind of continuous distortion of an "ambient space", 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 ''N'' and ''M'' be manifolds and ''g'' and ''h'' be [[embedding]]s of ''N'' in ''M''. A [[continuous map]]
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| :<math>F:M \times [0,1] \rightarrow M </math>
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| is defined to be an ambient isotopy taking ''g'' to ''h'' if ''F<sub>0</sub>'' is the [[identity function|identity map]], each map ''F<sub>t</sub>'' is a [[homeomorphism]] from ''M'' to itself, and ''F<sub>1</sub>'' ∘ ''g'' = ''h''. This implies that the [[orientation (geometry)|orientation]] must be preserved by ambient isotopies. For example, two knots which are [[mirror image]]s of each other are in general not equivalent.
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| ==See also==
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| *[[Regular homotopy]]
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| *[[Regular isotopy]]
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| == References ==
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| *Armstrong, ''Basic Topology'', Springer-Verlag, 1983
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| [[Category:Topology]]
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| [[Category:Maps of manifolds]]
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| {{topology-stub}}
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Latest revision as of 16:36, 17 September 2014
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