Ambient isotopy explained

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

F:M x [ 0 , 1 ] → M is defined to be an ambient isotopy taking g to h if F _{0} is the identity map, each map F _{t} is a homeomorphism from M to itself, and 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

• Isotopy
• Regular homotopy
• Regular isotopy

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