Ambient isotopy

Short description: Concept in toplogy

In [math]\displaystyle{ \mathbb{R}^3 }[/math], 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 [math]\displaystyle{ \mathbb{R}^4 }[/math].

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 [math]\displaystyle{ N }[/math] and [math]\displaystyle{ M }[/math] be manifolds and [math]\displaystyle{ g }[/math] and [math]\displaystyle{ h }[/math] be embeddings of [math]\displaystyle{ N }[/math] in [math]\displaystyle{ M }[/math]. A continuous map

[math]\displaystyle{ F:M \times [0,1] \rightarrow M }[/math]

is defined to be an ambient isotopy taking [math]\displaystyle{ g }[/math] to [math]\displaystyle{ h }[/math] if [math]\displaystyle{ F_0 }[/math] is the identity map, each map [math]\displaystyle{ F_t }[/math] is a homeomorphism from [math]\displaystyle{ M }[/math] to itself, and [math]\displaystyle{ F_1 \circ g = h }[/math]. 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