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

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

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.

is defined to be an ambient isotopy taking g to h if F₀ is the identity map, each map F_t is a homeomorphism from M to itself, and F₁ ∘ 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

• Regular homotopy
• Regular isotopy

References

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