Brunnian link

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In knot theory, a branch of mathematics, a Brunnian link is a nontrivial link that becomes trivial if any 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.

Contents 1 Brunnian braids 2 Real world examples 3 References 4 External links

[edit] Brunnian braids

A Brunnian braid is a braid which becomes trivial upon removal of any one of its strings. Brunnian braids form a subgroup of the braid group. Brunnian braids over the 2-sphere that are not Brunnian over the 2-disk 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 Hopf fibration S³ → S².

[edit] Real world examples

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

[edit] References

• Berrick, A. J.; Cohen, Frederick R.; Wong, Yan Loi & Wu, Jie (2006), “Configurations, braids, and homotopy groups”, Journal of the American Mathematical Society 19 (2): 265-326, MR2188127, <http://www.ams.org/jams/2006-19-02/S0894-0347-05-00507-2/> .
• Hermann Brunn, "Über Verkettung", J. Münch. Ber, XXII. 77-99 (1892). JFM 24.0507.01
• Dale Rolfsen (1976). Knots and Links. Berkeley: Publish or Perish, Inc. ISBN 0-914098-16-0.

[edit] External links

• "Are Borromean Links so Rare?", by Slavik Jablan (also available in its original form as published in the journal Forma here (PDF file)).