Brunnian link
In
The name Brunnian is after Hermann Brunn. Brunn's 1892 article Über Verkettung included examples of such links.
Examples
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:
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12-crossing link.
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18-crossing link.
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24-crossing link.
The simplest Brunnian link other than the 6-crossing Borromean rings is presumably the 10-crossing L10a140 link.[1]
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.[2]
In 2020, new and much more complicated Brunnian links have been discovered in [3] using highly flexible geometric-topology methods, far more than having been previously constructed. See Section 6.[3]
Non-circularity
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
Classification
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
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
In 2021, two special satellite operations were investigated for Brunnian links in 3-sphere, called "satellite-sum" and "satellite-tie", both of which can be used to construct infinitely many distinct Brunnian links from almost every Brunnian link.[5] A geometric classification theorem for Brunnian links was given.[5] More interestingly, a canonical geometric decomposition in terms of satellite-sum and satellite-tie, which is simpler than JSJ-decomposition, for Brunnian links, was developed. The building blocks of Brunnian links therein turn out to be Hopf -links, hyperbolic Brunnian links, and hyperbolic Brunnian links in unlink-complements, the last of which can be further reduced into a Brunnian link in 3-sphere.[5]
Massey products
Brunnian links can be understood in algebraic topology via Massey products: 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
A Brunnian
Real-world examples
Many
Brunnian chains are also used to create wearable and decorative items out of elastic bands using devices such as the Rainbow Loom or Wonder Loom.
References
- ^ Bar-Natan, Dror (2010-08-16). "All Brunnians, Maybe", [Academic Pensieve].
- ^ "Rubberband" Brunnian Links
- ^ ISSN 0218-2165.
- ; see in particular Lemma 3.2, p. 89
- ^ ISSN 0218-2165.
Further reading
- Berrick, A. Jon; Cohen, Frederick R.; Wong, Yan Loi; Wu, Jie (2006), "Configurations, braids, and homotopy groups", MR 2188127.
- Hermann Brunn, "Über Verkettung", J. Münch. Ber, XXII. 77–99 (1892). JFM 24.0507.01(in German)
- JSTOR 1969685
- Rolfsen, Dale (1976), Knots and Links, Mathematics Lecture Series, vol. 7, MR 0515288
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) Archived 2021-02-28 at the Wayback Machine).
- "Brunnian_link", The Knot Atlas.