1.2  Shadows of KLs 

Knots have been used by mankind from prehistoric times up to ancient periods of European and Egyptian civilizations, even serving as the basis for mathematical recording systems (e.g., for Inca quipu). Examples can be found in all civilizations, in Chinese art, Celtic art, ethnic Tamil art and Tchokwe art, in Arabian, Greek or Smyrnian laces... In modern science and art, KLs are to be found in DNA, in chemistry, in physics, in nature, in sculpture, etc. From the design point of view, they are an example of modular structures (Jablan, 2002), because they can be composed from only five basic pieces (modules). 

The main difference between "real" and "mathematical" KLs is that the first are open-ended, and the others are closed. 

Before giving the precise definition of KLs, we will start with the intuitive notion of knotting. Intuitively speaking, a mathematical knot is a knotted loop of string where the two ends of the string are glued together. The result is a string (having no thickness) that has no loose ends and that is truly knotted. A link is a set of such knotted loops, all tangled up together. A knot is then a closed curve in space that does not intersect itself anywhere, and a link is a collection of several such mutually disjoint curves (Adams, 1994). 

In the language of mathematics, a knot is a smooth or piecewise linear embedding of a circle S1 into Euclidean 3-dimensional space Â3 (or the 3-dimensional sphere S3), and a c-component link is an embedding of c disjoint copies of a circle S1 into Â3 (or S3), where the circles Si1 are its components (i = 1,2,...,c). Because a knot is just a 1-component link (c = 1), except in cases that some property is restricted only to knots, we will use the more general term "link" or KL (knot or link) for all of them. Knot theory studies the placement problem: it tries to classify how the set {Si1} (i = 1,2,...,c) is placed in Â3 (or S3). 

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