Ambient isotopy

{{Short description|Concept in toplogy}}

{{multiple image

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| image1 = Blue Unknot.png

| image2 = Blue Trefoil Knot.png

| footer = In $\backslash mathbb\{R\}^3$, 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 $\backslash mathbb\{R\}^4$.

}}

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 $N$ and $M$ be manifolds and $g$ and $h$ be embeddings of $N$ in $M$. A continuous map

:$F:M\; \backslash times\; [0,1]\; \backslash rightarrow\; M$

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 $F\_1\; \backslash circ\; g\; =\; h$. 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.