How to braid 2 strings 2018

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This four-component link strings is a Brunnian link. Six-component Brunnian link.

In, a branch of, a Brunnian link is a nontrivial that becomes a set of trivial 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 ).

The name Brunnian is after. Brunn's 1892 article Über Verkettung included examples of such links.



The best-known and simplest possible Brunnian link is the, a link of three. 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.

An example of a n-component Brunnian link is given by the, 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 by in (), 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 – which in this case (but not in general) is the of the – 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 on n generators, Fn, as the link group of a single link is the of the, which is the integers, and the link group of an unlinked union is the of the link groups of the components.

Not every element of the link group gives a Brunnian link, as removing any other component must also unlink the remaining n elements. Milnor showed that the group elements that do correspond to Brunnian links are related to the of the of the free group, which can be interpreted as "relations" in the.

Massey products[]

Main article:

Brunnian links can be understood in via : a Massey product is an n-fold product which is only defined if all (n − 1)-fold products of its terms vanish. This corresponds to the Brunnian property of all (n − 1)-component sublinks being unlinked, but the overall n-component link being non-trivially linked.

Brunnian braids[]

See also:

The standard braid is Brunnian: if one removes the black strand, the blue strand is always on top of the red strand, and they are thus not braided around each other; likewise for removing other strands.

A Brunnian is a braid that becomes trivial upon removal of any one of its strings. Brunnian braids form a of the. Brunnian braids over the 2- that are not Brunnian over the 2- give rise to non-trivial elements in the homotopy groups of the 2-sphere. For example, the "standard" braid corresponding to the Borromean rings gives rise to the S3 → S2, and iterations of this (as in everyday braiding) is likewise Brunnian.

Real-world examples[]

Many and some are variants of Brunnian Links, with the goal being to free a single piece only partially linked to the rest, thus dismantling the structure.

Brunnian chains are also used to create wearable and decorative items out of elastic bands using devices such as the or.


Further reading[]

  • Berrick, A. Jon; Cohen, Frederick R.; Wong, Yan Loi; Wu, Jie (2006),,, 19 (2): 265–326, :,   .
  • Hermann Brunn, "Über Verkettung", J. Münch. Ber, XXII. 77–99 (1892).   (in German)
  • (March 1954), "Link Groups",, Annals of Mathematics, 59 (2): 177–195, :,   
  • Rolfsen, Dale (1976),, Mathematics Lecture Series, 7, : Publish or Perish,  ,   

External links[]

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