Mathematical Recreations: Knotting Ventured...; July 2000; Scientific American Magazine; by Stewart; 2 Page(s)

In the past century the study of knots has become a major area of mathematical research. Knots embody one of the big questions in topology: What are the different ways to position one geometric form inside another? In the case of knots, the two forms are a circle-which can be represented by a closed loop of string-and the whole of three-dimensional space. As far as topologists are concerned, a knot is a circle that has been embedded in three-dimensional space in such a manner that it cannot be disentangled by continuously deforming the space around it.

This description is somewhat removed from everyday experience: in the real world, bits of string have ends, and when you try to untie a knot you deform the string, not the space around it. Although the topological definition captures the "knottiness" of knots, other aspects do not reduce so well to a topological formulation. A clear case in point is the problem of knotting two pieces of string together to form a single, longer piece. The main requirement is that the knot should not slip if you pull on the ends of the string. Surface friction and the material from which the string is made come into play, so the task requires a different approach.