The fact that every knot represents a homeomorphic image of a circle, and a link is a homeomorphic image of c disjoint circles is not sufficient to compare them: we are interested in their embedding in the space ³, and not just in the homeomorphism. In this case we can try another possibility: to compare their complements with regard to the space ³. It is easy to see that if two links L₁ and L₂ are ambient isotopic, their complements are homeomorphic. Also, a link and its mirror image have homeomorphic complements. What will happen with inverse statements? For knots is true that two knots K₁ and K₂ are ambient isotopic iff their complements are homeomorphic (Gordon and Luecke, 1989). However, that statement is not true for links; according to a famous Whitehead example, two links that are not ambient isotopic can have homeomorphic complements. Whitehead proved that there are infinitely many links with the same complement as the Whitehead link (Whitehead, 1937; Gordon, 2002).

In order to avoid the necessity of introducing differentiable or smooth curves, as well to avoid some peculiar cases such as wild KLs, we can think about KLs as piecewise linear. A link L is called tame if it is ambient isotopic to a collection of simple closed polygons in ³. Otherwise, it is wild. A link consisting of c closed polygonal lines in ³ will be called a polygonal link. For polygonal KLs an elementary isotopy is achieved either by subdividing an edge AB by the vertex C, or by applying a contraction on AC and CB. An ambient isotopy for a polygonal KL is a finite sequence of elementary isotopies.