A Brunnian link is a nontrivial link in a collection of links which causes the entire collective to fall apart if it is cut. A well known example of a Brunnian link are the Borromean rings, three rings which are connected in such a way that the removal of one ring would cause the whole thing to separate. Brunnian links can potentially link an infinite number of items, and they are a topic of immense interest in knot theory, a branch of mathematics. While knot theory might not sound terribly scintillating, it is actually a very interesting branch of the mathematics field.
The Brunnian link is named for Hermann Brunn, a 19th century mathematician who wrote about the phenomenon and covered it in a paper. In addition to just being interesting, Brunnian links can also have practical and theoretical applications. Molecular biologists, for example, have worked with Brunnian links to model various physical structures. Some people have also made a study of Brunnian braids, a closely related concept.
In things like Borromean rings, the individual links are unknots, closed loops formed without any knots. The most obvious example of an unknot is a simple loop, like a ring, but unknots can also get extremely complex, and it is possible create amazingly ornate structures of Brunnian links with unknots. The Brunnian link illustrates the importance that a simple object or action can have, which is why Borromean rings are often used to symbolize strength in unity.
Specialists in three dimensional modeling have produced some very intriguing and complex working models of Brunnian links which easily illustrate the principle without the need of a physical example. Such models are usually designed to allow users to manipulate them for several different angle views, and it is possible to remove a link to see an illustration of the Brunnian link in action.
You may be more familiar with Brunnian links than you know. These links often play a vital role in brain teasers which require their users to physically disentangle several items. When the user finds the correct method for manipulating the physical puzzle, he or she can cause it to fall apart, and then the next challenge is reassembling it. Puzzle rings are another well known example of the Brunnian link, as most are designed in such a way that if a ring is removed, the entire ring falls apart.