Brunnian link
In knot theory, a branch of topology, a Brunnian link is a nontrivial link that becomes a set of trivial unlinked circles if any one component is removed. In other words, cutting any loop frees all the other loops (so that no two loops can be directly linked).
The name Brunnian is after Hermann Brunn. Brunn's 1892 article Über Verkettung included examples of such links.
The best-known and simplest possible Brunnian link is the Borromean rings, a link of three unknots. However for every number three or above, there are an infinite number of links with the Brunnian property containing that number of loops. Here are some relatively simple three-component Brunnian links which are not the same as the Borromean rings:
12-crossing link.
18-crossing link.
24-crossing link.
The simplest Brunnian link other than the 6-crossing Borromean rings is presumably the 10-crossing L10a140 link.
An example of a n-component Brunnian link is given by the "Rubberband" Brunnian Links, where each component is looped around the next as aba−1b−1, with the last looping around the first, forming a circle.
Brunnian links were classified up to link-homotopy by John Milnor in (Milnor 1954), and the invariants he introduced are now called Milnor invariants.
An (n + 1)-component Brunnian link can be thought of as an element of the link group – which in this case (but not in general) is the fundamental group of the link complement – of the n-component unlink, since by Brunnianness removing the last link unlinks the others. The link group of the n-component unlink is the free group on n generators, Fn, as the link group of a single link is the knot group of the unknot, which is the integers, and the link group of an unlinked union is the free product of the link groups of the components.
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