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
 * $$F:M \times [0,1] \rightarrow M $$

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 $$F_1 \circ g = h$$. 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.