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 the ambient
isotopy is equivalent to allowing that threads can be twisted and moved in a continuous way in space (cutting and gluing back together is not allowed). Usually, one says ``knot'' or ``link''
meaning "tame knot" or "tame link". Otherwise, one deals with a wild knot or link.  In order to describe ambient isotopy in the language of mathematics, we need some topological background. 

Let X be a non-empty set. A topology on X is a collection of subsets of X, called open sets, satisfying: 

  1. the empty set and X itself are open sets; 
  2. an arbitrary union of open sets is an open set; 
  3. the intersection of finitely many open sets is an open set.
The set X together with a collection of open sets satisfying the rules (1), (2), (3) is a topological space. 

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: 

  1. h0 = H((x,y,x),0) is the identity Â3®Â3, 
  2. for all t Î [0,1], ht = H((x,y,z),t) is a homeomorphism Â3®Â3, 
  3. if h1 = H((x,y,z),1), then h1(K) = K'. 
If K and K1 are ambient-isotopic, then the knots ht(K) (t Î [0,1]) form a continuously varying family deforming K into K1. That definition can be extended to links in a natural way (Gilbert and Porter, 1994). 
 
 

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