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

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 Ft is a homeomorphism from M to itself, and F1g = 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


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