Ambient isotopy
# Ambient isotopy

$\mathbb {R} ^{3}$

$\mathbb {R} ^{4}$

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.

