# Ambient isotopy

In the mathematical subject of topology, an **ambient isotopy**, also called an *h-isotopy*, is a kind of continuous distortion of an ambient space, for example 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

is defined to be an ambient isotopy taking *g* to *h* if *F*_{0} is the identity map, each map *F _{t}* is a homeomorphism from

*M*to itself, and

*F*

_{1}∘

*g*=

*h*. This implies that the orientation must be preserved by ambient isotopies. For example, two knots that are mirror images of each other are in general not equivalent.

## See also

## References

- M. A. Armstrong,
*Basic Topology*, Springer-Verlag, 1983