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", 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". The map $F\_t:M\; imes\; [0,1]\; ightarrow\; M$ is defined to be an ambient isotopy taking "g" to "h" if $F\_0$ is "g", $F\_1$ is "h" and $F\_t$ induces and in addition, every $t\; in\; [0,1]$ must induce a self homeomorphism of M. This implies that the orientation must be preserved by ambient isotopies. For example, two knots mirror image of each other are in general not equivalent.

ee also

*regular homotopy
*regular isotopy

References

*Armstrong, "Basic Topology", Springer-Verlag, 1983