Brunnian link: Difference between revisions
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[[Image:Brunnian.gif|right|200px|thumb|This four-component link is a Brunnian link.]] |
[[Image:Brunnian.gif|right|200px|thumb|This four-component link is a Brunnian link.]] |
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[[Image:6Loops-Brunnian-link.png|right|200px|thumb|Six-component Brunnian link.]] |
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A '''Brunnian link''' is a nontrivial link which becomes trivial if any component is removed. In other words, cutting any loop frees all the other loops (so that there is no [[L2a1|Hopf link]] between any two loops). |
A '''Brunnian link''' is a nontrivial link which becomes trivial if any component is removed. In other words, cutting any loop frees all the other loops (so that there is no [[L2a1|Hopf link]] between any two loops). |
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The name Brunnian is after Hermann Brunn, whose 1892 article '' |
The name Brunnian is after Hermann Brunn, whose 1892 article ''Über Verkettung'' included examples of such links. |
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The best-known and simplest possible Brunnian link is the [[L6a4|Borromean rings]], a link of three unknots. (This is also the only Brunnian link |
The best-known and simplest possible Brunnian link is the [[L6a4|Borromean rings]], a link of three unknots. (This is also the only Brunnian link with less than 10 crossings, and the only symmetrical Brunnian link with less than 12 crossings.) 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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<gallery> |
<gallery> |
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Image:Brunnian-3-not-Borromean.gif|12-crossing link. |
Image:Brunnian-3-not-Borromean.gif|12-crossing link. |
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Image:Three-triang-18crossings-Brunnian.gif|18-crossing link. |
Image:Three-triang-18crossings-Brunnian.gif|18-crossing link. |
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Image:Three-squares-Brunnian-24crossings.png|24-crossing link. |
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Image:Three-interlaced-pentagons-Brunnian-30crossings.png|30-crossing link. |
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</gallery> |
</gallery> |
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T. Kanenobu has proven there exist hyperbolic Brunnian links with arbitrarily many components. |
T. Kanenobu has proven there exist hyperbolic Brunnian links with arbitrarily many components. |
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==Non-Borromean-rings Brunnian links in real world symbols== |
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[[Image:Selo Rito Brasileiro.jpg|thumb|Brazilian Masonic emblem with three interlaced triangles]] |
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A version of the three interlaced triangles (18-crossings Brunnian link) is also found in the sixth century Marundheeswarar Temple in Thiruvanmayur, South Chennai, India, associated with the goddess Tripurasundari,[http://www.liv.ac.uk/%7Espmr02/rings/maths.html] and is occasionally called "the star of the muses" as a minor symbol in some New Age / occultist / neo-pagan circles. |
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==See also== |
==See also== |
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*[[L10a140]] |
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*[[L11a434]] |
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*[[L11n436]] |
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*[["Rubberband" Brunnian Links]] |
*[["Rubberband" Brunnian Links]] |
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==Further reading== |
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T.Kanenobu, Hyperbolic links with Brunnian properties. J. Math. Soc. Japan. Vol 38 No. 2, (1986) 295--308. |
T.Kanenobu, Hyperbolic links with Brunnian properties. J. Math. Soc. Japan. Vol 38 No. 2, (1986) 295--308. |
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==Links== |
==Links== |
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*[http://en.wikipedia.org/wiki/Brunnian_link Wikipedia article "Brunnian link"] |
*[http://en.wikipedia.org/wiki/Brunnian_link Wikipedia article "Brunnian link"] |
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*[http://www.mi.sanu.ac.yu/vismath/bor/bor1.htm "Are Borromean Links so Rare?", by Slavik Jablan] |
*[http://www.mi.sanu.ac.yu/vismath/bor/bor1.htm "Are Borromean Links so Rare?", by Slavik Jablan] ([http://www.scipress.org/journals/forma/abstract/1404/14040269.html Original journal site]) |
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Latest revision as of 13:28, 29 April 2013
A Brunnian link is a nontrivial link which becomes trivial if any component is removed. In other words, cutting any loop frees all the other loops (so that there is no Hopf link between any two loops).
The name Brunnian is after Hermann Brunn, whose 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. (This is also the only Brunnian link with less than 10 crossings, and the only symmetrical Brunnian link with less than 12 crossings.) 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.
30-crossing link.
T. Kanenobu has proven there exist hyperbolic Brunnian links with arbitrarily many components.
Non-Borromean-rings Brunnian links in real world symbols
A version of the three interlaced triangles (18-crossings Brunnian link) is also found in the sixth century Marundheeswarar Temple in Thiruvanmayur, South Chennai, India, associated with the goddess Tripurasundari,[1] and is occasionally called "the star of the muses" as a minor symbol in some New Age / occultist / neo-pagan circles.
See also
Further reading
T.Kanenobu, Hyperbolic links with Brunnian properties. J. Math. Soc. Japan. Vol 38 No. 2, (1986) 295--308.
