Ambient isotopy

In $\mathbb {R} ^{3}$, 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. They are ambient-isotopic in $\mathbb {R} ^{4}$.

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.

See also

• Isotopy
• Regular homotopy
• Regular isotopy

References

• M. A. Armstrong, Basic Topology, Springer-Verlag, 1983
• Sasho Kalajdzievski, An Illustrated Introduction to Topology and Homotopy, CRC Press, 2010, Chapter 10: Isotopy and Homotopy