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Ambient isotopy

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Concept in toplogy In R 3 {\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 R 4 {\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 N {\displaystyle N} and M {\displaystyle M} be manifolds and g {\displaystyle g} and h {\displaystyle h} be embeddings of N {\displaystyle N} in M {\displaystyle M} . A continuous map

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

is defined to be an ambient isotopy taking g {\displaystyle g} to h {\displaystyle h} if F 0 {\displaystyle F_{0}} is the identity map, each map F t {\displaystyle F_{t}} is a homeomorphism from M {\displaystyle M} to itself, and F 1 ∘ g = h {\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

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


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