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 and be manifolds and and be embeddings of in . A continuous map

In , 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 .

is defined to be an ambient isotopy taking to if is the identity map, each map is a homeomorphism from to itself, and . 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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References

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  • 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