Ambient isotopy

The unknot is not ambient-isotopic to the trefoil knot since one cannot be deformed into the other through a continuous path of homeomorphisms of the ambient space.

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.

References

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

Ambient isotopy

See also

References