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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]] | |||
:<math>F:M \times [0,1] \rightarrow M </math> | |||
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. | |||
==See also== | |||
*[[Regular homotopy]] | |||
*[[Regular isotopy]] | |||
== References == | |||
*Armstrong, ''Basic Topology'', Springer-Verlag, 1983 | |||
[[Category:Topology]] | |||
[[Category:Maps of manifolds]] | |||
{{topology-stub}} | |||
Revision as of 21:46, 3 February 2014
In the 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 knots 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 embeddings of N in M. A continuous map
![{\displaystyle F:M\times [0,1]\rightarrow M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c08f131a634efc22cb9287fd2e47dd913cc2b84)
is defined to be an ambient isotopy taking g to h if F0 is the identity map, each map Ft is a homeomorphism from M to itself, and F1 ∘ g = h. This implies that the orientation must be preserved by ambient isotopies. For example, two knots which are mirror images of each other are in general not equivalent.
See also
References
- Armstrong, Basic Topology, Springer-Verlag, 1983