Ambient isotopy

In ${\displaystyle \mathbb {R} ^{3}}$, 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 ${\displaystyle \mathbb {R} ^{4}}$.

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 ${\displaystyle N}$ and ${\displaystyle M}$ be manifolds and ${\displaystyle g}$ and ${\displaystyle h}$ be embeddings of ${\displaystyle N}$ in ${\displaystyle M}$. A continuous map

${\displaystyle F:M\times [0,1]\rightarrow M}$

is defined to be an ambient isotopy taking ${\displaystyle g}$ to ${\displaystyle h}$ if each ${\displaystyle F_{t}:M\rightarrow M,F_{t}(\cdot )=F(\cdot ,t)}$ is a homeomorphism from ${\displaystyle M}$ to itself, ${\displaystyle F_{0}}$ is the identity map and ${\displaystyle F_{1}\circ g=h}$. 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