Chapter 5: Links
Another sort of tangling is linking. In some ways this can be simpler than knotting, but it requires at least two strings. We can have the same questions with open versus closed loops in linking that we had with knotting, but with linking the most common everyday links are with closed loops. For example here is the simplest link.
This sort of link is very common — links made up of linked simple closed loops — closed loops without knots in them — are keychains, and chains of all sorts, from necklace and bracelet chains to heavy chains used for ship anchors or anchoring heavy machinery.
The pattern of linking matters. For example, to the right we have links of two simple loops, but with different linking patterns. The closed loops in a link are called components. Both of these patterns have two components.
Links include patterns where one or more of the components are knotted, as we see here. In this sense links are a larger category than knots: knots are one component links.
Linking is more subtle than it may first appear. There is a link of three loops, which cannot come apart, but if you look at any pair of loops, they are not linked with one another. Therefore, if you cut just one of the loops, it all comes apart. It is usually called the Borromean Rings, after an aristocratic family who used it in their family crest, but it was known at least as far back as the 6th and 7th century to the Vikings, in India and Japan. On the right is the usual version, below it a more complex pattern with the same properties.
