Two links L and L_{1}
are ambient isotopic iff there is a continuous movement (or deformation)
of a link L through 3D space, without letting it pass through itself,
that transforms L into L_{1}.
In an ambient isotopy,
it is not allowed to shrink a part of the link down to a point. If we imagine
that the curves defining a link are a fine thread, flexible and elastic, then
the ambient Let X be a nonempty set. A topology on X is a collection of subsets of X, called open sets, satisfying:
Let X and Y be topological spaces. A function f: X® Y is continuous if, for every open set V in Y, its original f^{1}(V) is open in X. Let X and Y be topological spaces. A function f: X® Y which is bijective, continuous, and has a continuous inverse f^{1}: Y ® X is called a homeomorphism. A function f: X® Y which is injective, continuous, and such that the bijection f: X ® f(X) has a continuous inverse is called an embedding. In the intuitive equality criterion for links L and L_{1} described before, we think of moving their images around Â^{3} in a continuous manner until one coincides with the other. If the parameter t represents the time, that definition provides a continuous sequence of homeomorphisms of Â^{3} from time t = 0 to t = 1, as a kind of continuous movie. From that follows the mathematical definition of ambient isotopy for knots K and K_{1}: K and K_{1} are ambientisotopic if there exists a continuous function H: Â^{3}×[0,1]®Â^{3} such that:
