|Two links L and L1
are ambient isotopic iff there is a continuous movement (or deformation)
of a link L through 3-D space, without letting it pass through itself,
that transforms L into L1.
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
Let X be a non-empty 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 L1 described before, we think of moving their images around Â3 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 Â3 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 K1: K and K1 are ambient-isotopic if there exists a continuous function H: Â3×[0,1]®Â3 such that: