Ambient Isotopy

An ambient isotopy from an embedding of a manifold  in  to another is a homotopy of self diffeomorphisms (or isomorphisms, or piecewise-linear transformations, etc.) of , starting at the identity map, such that the "last" diffeomorphism compounded with the first embedding of  is the second embedding of . In other words, an ambient isotopy is like an isotopy except that instead of distorting the embedding, the whole ambient space is being stretched and distorted and the embedding is just "coming along for the ride." For smooth manifolds, a map is isotopic iff it is ambiently isotopic.

For knots, the equivalence of manifolds under continuous deformation is independent of the embedding space. Knots of opposite chirality have ambient isotopy, but not regular isotopy.

REFERENCES:

Hirsch, M. W. Differential Topology. New York: Springer-Verlag, 1988.

Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First  Knots." Math. Intell. 20, 33-48, Fall 1998.