Ambient isotopy

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In the mathematical subject of topology, an ambient isotopy, also called an h-isotopy, is a kind of continuous distortion of an "ambient space", 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 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 which are mirror images of each other are in general not equivalent.

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References[edit]

  • Armstrong, Basic Topology, Springer-Verlag, 1983