Isotopy

An isotopy is a continuous one-parameter family of embeddings or homeomorphisms beginning at a given map.

Two embeddings $f_0,f_1:N\hookrightarrow M$ are ambient isotopic when a continuously varying family of ambient homeomorphisms $F_t$ starting at the identity satisfies $F_1{f}_0=f_1$. This is stronger than an arbitrary homotopy of maps. Smooth versions use smooth families of diffeomorphisms; unparametrized versions identify embedded images, and oriented versions preserve their orientations.

A smooth isotopy is a smooth family of smooth embeddings or diffeomorphisms parametrized by an interval and beginning at the given map.