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Concept
Brunnian link
Summary
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:
Image:Brunnian-3-not-Borromean.svg|12-crossing link.
Image:Three-triang-18crossings-Brunnian.svg|18-crossing link.
Image:Three-interlaced-squares-Brunnian-24crossings.svg|24-crossing link.
The simplest Brunnian link other than the 6-crossing Borromean rings is presumably the 10-crossing L10a140 link.
An example of an 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.
In 2020, new and much more complicated Brunnian links have been discovered in using highly flexible geometric-topology methods, far more than having been previously constructed. See Section 6.
It is impossible for a Brunnian link to be constructed from geometric circles. Somewhat more generally, if a link has the property that each component is a circle and no two components are linked, then it is trivial. The proof, by Michael Freedman and Richard Skora, embeds the three-dimensional space containing the link as the boundary of a Poincaré ball model of four-dimensional hyperbolic space, and considers the hyperbolic convex hulls of the circles. These are two-dimensional subspaces of the hyperbolic space, and their intersection patterns reflect the pairwise linking of the circles: if two circles are linked, then their hulls have a point of intersection, but with the assumption that pairs of circles are unlinked, the hulls are disjoint.
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