Brunnian link - Revision history

78.51.208.159: we can english

2024-09-09T11:25:17Z

we can english

← Previous revision		Revision as of 11:25, 9 September 2024
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	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''<sup>−1</sup>''b''<sup>−1</sup>, with the last looping around the first, forming a circle.<ref>[http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links "Rubberband" Brunnian Links]</ref>		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''<sup>−1</sup>''b''<sup>−1</sup>, with the last looping around the first, forming a circle.<ref>[http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links "Rubberband" Brunnian Links]</ref>

	In 2020, new and much more complicated Brunnian links ~~have been~~ discovered in <ref name=":1">{{Cite journal \|last=Bai \|first=Sheng \|last2=Wang \|first2=Weibiao \|date=November 2020 \|title=New criteria and constructions of Brunnian links \|url=https://www.worldscientific.com/doi/abs/10.1142/S0218216520430087 \|journal=Journal of Knot Theory and Its Ramifications \|volume=29 \|issue=13 \|pages=2043008 \|doi=10.1142/S0218216520430087 \|issn=0218-2165\|arxiv=2006.10290 }}</ref> using highly flexible geometric-topology methods~~, far more than having been previously constructed~~. See Section 6.<ref name=":1" />		In 2020, new and much more complicated Brunnian links were discovered in <ref name=":1">{{Cite journal \|last=Bai \|first=Sheng \|last2=Wang \|first2=Weibiao \|date=November 2020 \|title=New criteria and constructions of Brunnian links \|url=https://www.worldscientific.com/doi/abs/10.1142/S0218216520430087 \|journal=Journal of Knot Theory and Its Ramifications \|volume=29 \|issue=13 \|pages=2043008 \|doi=10.1142/S0218216520430087 \|issn=0218-2165\|arxiv=2006.10290 }}</ref> using highly flexible geometric-topology methods. See Section 6.<ref name=":1" />

	==Non-circularity==		==Non-circularity==

77.12.211.111: spelling

2024-05-29T20:18:04Z

spelling

← Previous revision		Revision as of 20:18, 29 May 2024
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	==Real-world examples==		==Real-world examples==
	[[File:Rainbow Loom wide bracelet.jpg\|thumb\|Rainbow loom bracelet showing ~~Brunian~~ chains]]		[[File:Rainbow Loom wide bracelet.jpg\|thumb\|Rainbow loom bracelet showing Brunnian chains]]
	Many [[disentanglement puzzle]]s 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.		Many [[disentanglement puzzle]]s 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.

77.12.211.111: /* Brunnian braids */

2024-05-29T20:17:26Z

Brunnian braids

← Previous revision		Revision as of 20:17, 29 May 2024
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	{{see also\|Closed braid}}		{{see also\|Closed braid}}
	[[File:Braid StepBystep.jpg\|thumb\|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.]]		[[File:Braid StepBystep.jpg\|thumb\|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 [[Braid theory\|braid]] is a braid that 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 (mathematics)\|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]] ''S''<sup>3</sup> → ''S''<sup>2</sup>, and ~~iterations~~ of this (as in everyday braiding) is likewise Brunnian.		A Brunnian [[Braid theory\|braid]] is a braid that 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 (mathematics)\|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]] ''S''<sup>3</sup> → ''S''<sup>2</sup>, and iteration of this (as in everyday braiding) is likewise Brunnian.

	==Real-world examples==		==Real-world examples==

InternetArchiveBot: Rescuing 1 sources and tagging 0 as dead.) #IABot (v2.0.9.5

2023-10-29T20:17:28Z

Rescuing 1 sources and tagging 0 as dead.) #IABot (v2.0.9.5

← Previous revision		Revision as of 20:17, 29 October 2023
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	==External links==		==External links==
	* [http://www.mi.sanu.ac.rs/vismath/bor/bor1.htm "Are Borromean Links so Rare?", by Slavik Jablan] (also available in its original form as published in the journal ''Forma'' [http://www.scipress.org/journals/forma/pdf/1404/14040269.pdf here (PDF file)]).		* [http://www.mi.sanu.ac.rs/vismath/bor/bor1.htm "Are Borromean Links so Rare?", by Slavik Jablan] (also available in its original form as published in the journal ''Forma'' [http://www.scipress.org/journals/forma/pdf/1404/14040269.pdf here (PDF file)] {{Webarchive\|url=https://web.archive.org/web/20210228100158/http://www.scipress.org/journals/forma/pdf/1404/14040269.pdf \|date=2021-02-28 }}).
	* {{Knot Atlas\|Brunnian_link}}		* {{Knot Atlas\|Brunnian_link}}

OAbot: Open access bot: arxiv added to citation with #oabot.

2023-08-12T13:54:46Z

Open access bot: arxiv added to citation with #oabot.

← Previous revision		Revision as of 13:54, 12 August 2023
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	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''<sup>−1</sup>''b''<sup>−1</sup>, with the last looping around the first, forming a circle.<ref>[http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links "Rubberband" Brunnian Links]</ref>		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''<sup>−1</sup>''b''<sup>−1</sup>, with the last looping around the first, forming a circle.<ref>[http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links "Rubberband" Brunnian Links]</ref>

	In 2020, new and much more complicated Brunnian links have been discovered in <ref name=":1">{{Cite journal \|last=Bai \|first=Sheng \|last2=Wang \|first2=Weibiao \|date=November 2020 \|title=New criteria and constructions of Brunnian links \|url=https://www.worldscientific.com/doi/abs/10.1142/S0218216520430087 \|journal=Journal of Knot Theory and Its Ramifications \|volume=29 \|issue=13 \|pages=2043008 \|doi=10.1142/S0218216520430087 \|issn=0218-2165}}</ref> using highly flexible geometric-topology methods, far more than having been previously constructed. See Section 6.<ref name=":1" />		In 2020, new and much more complicated Brunnian links have been discovered in <ref name=":1">{{Cite journal \|last=Bai \|first=Sheng \|last2=Wang \|first2=Weibiao \|date=November 2020 \|title=New criteria and constructions of Brunnian links \|url=https://www.worldscientific.com/doi/abs/10.1142/S0218216520430087 \|journal=Journal of Knot Theory and Its Ramifications \|volume=29 \|issue=13 \|pages=2043008 \|doi=10.1142/S0218216520430087 \|issn=0218-2165\|arxiv=2006.10290 }}</ref> using highly flexible geometric-topology methods, far more than having been previously constructed. See Section 6.<ref name=":1" />

	==Non-circularity==		==Non-circularity==
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	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 [[graded Lie algebra]] of the [[lower central series]] of the free group, which can be interpreted as "relations" in the [[free Lie algebra]].		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 [[graded Lie algebra]] of the [[lower central series]] of the free group, which can be interpreted as "relations" in the [[free Lie algebra]].

	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.<ref name=":0" /> A geometric classification theorem for Brunnian links was given.<ref name=":0" /> 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.<ref name=":0">{{Cite journal \|last=Bai \|first=Sheng \|last2=Ma \|first2=Jiming \|date=September 2021 \|title=Satellite constructions and geometric classification of Brunnian links \|url=https://www.worldscientific.com/doi/10.1142/S0218216521400058 \|journal=Journal of Knot Theory and Its Ramifications \|volume=30 \|issue=10 \|pages=2140005 \|doi=10.1142/S0218216521400058 \|issn=0218-2165}}</ref>		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.<ref name=":0" /> A geometric classification theorem for Brunnian links was given.<ref name=":0" /> 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.<ref name=":0">{{Cite journal \|last=Bai \|first=Sheng \|last2=Ma \|first2=Jiming \|date=September 2021 \|title=Satellite constructions and geometric classification of Brunnian links \|url=https://www.worldscientific.com/doi/10.1142/S0218216521400058 \|journal=Journal of Knot Theory and Its Ramifications \|volume=30 \|issue=10 \|pages=2140005 \|doi=10.1142/S0218216521400058 \|issn=0218-2165\|arxiv=1906.01253 }}</ref>

	=== Massey products ===		=== Massey products ===

WikiCleanerBot: v2.05b - Bot T20 CW#61 - Fix errors for CW project (Reference before punctuation)

2023-06-03T17:23:29Z

v2.05b - Bot T20 CW#61 - Fix errors for CW project (Reference before punctuation)

← Previous revision		Revision as of 17:23, 3 June 2023
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	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''<sup>−1</sup>''b''<sup>−1</sup>, with the last looping around the first, forming a circle.<ref>[http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links "Rubberband" Brunnian Links]</ref>		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''<sup>−1</sup>''b''<sup>−1</sup>, with the last looping around the first, forming a circle.<ref>[http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links "Rubberband" Brunnian Links]</ref>

	In 2020, new and much more complicated Brunnian links have been discovered in <ref name=":1">{{Cite journal \|last=Bai \|first=Sheng \|last2=Wang \|first2=Weibiao \|date=November 2020 \|title=New criteria and constructions of Brunnian links \|url=https://www.worldscientific.com/doi/abs/10.1142/S0218216520430087 \|journal=Journal of Knot Theory and Its Ramifications \|volume=29 \|issue=13 \|pages=2043008 \|doi=10.1142/S0218216520430087 \|issn=0218-2165}}</ref> using highly flexible geometric-topology methods, far more than having been previously constructed. See Section 6 <ref name=":1" />.		In 2020, new and much more complicated Brunnian links have been discovered in <ref name=":1">{{Cite journal \|last=Bai \|first=Sheng \|last2=Wang \|first2=Weibiao \|date=November 2020 \|title=New criteria and constructions of Brunnian links \|url=https://www.worldscientific.com/doi/abs/10.1142/S0218216520430087 \|journal=Journal of Knot Theory and Its Ramifications \|volume=29 \|issue=13 \|pages=2043008 \|doi=10.1142/S0218216520430087 \|issn=0218-2165}}</ref> using highly flexible geometric-topology methods, far more than having been previously constructed. See Section 6.<ref name=":1" />

	==Non-circularity==		==Non-circularity==

Noboru Kashiwa: /* Examples */

2023-05-31T09:21:34Z

Examples

← Previous revision		Revision as of 09:21, 31 May 2023
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	The simplest Brunnian link other than the 6-crossing Borromean rings is presumably the 10-crossing [[L10a140 link]].<ref>[[Dror Bar-Natan\|Bar-Natan, Dror]] (2010-08-16). "[http://drorbn.net/AcademicPensieve/2010-08/nb/All%20Brunnians,%20Maybe.pdf All Brunnians, Maybe]", ''[Academic Pensieve]''.</ref>		The simplest Brunnian link other than the 6-crossing Borromean rings is presumably the 10-crossing [[L10a140 link]].<ref>[[Dror Bar-Natan\|Bar-Natan, Dror]] (2010-08-16). "[http://drorbn.net/AcademicPensieve/2010-08/nb/All%20Brunnians,%20Maybe.pdf All Brunnians, Maybe]", ''[Academic Pensieve]''.</ref>

	An example of a ''n''-component Brunnian link is given by the "rubberband" Brunnian Links, where each component is looped around the next as ''aba''<sup>−1</sup>''b''<sup>−1</sup>, with the last looping around the first, forming a circle.<ref>[http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links "Rubberband" Brunnian Links]</ref>		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''<sup>−1</sup>''b''<sup>−1</sup>, with the last looping around the first, forming a circle.<ref>[http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links "Rubberband" Brunnian Links]</ref>

	In 2020, new and much more complicated Brunnian links have been discovered in <ref>{{Cite journal \|last=Bai \|first=Sheng \|last2=Wang \|first2=Weibiao \|date=November 2020 \|title=New criteria and constructions of Brunnian links \|url=https://www.worldscientific.com/doi/abs/10.1142/S0218216520430087 \|journal=Journal of Knot Theory and Its Ramifications \|volume=29 \|issue=13 \|pages=2043008 \|doi=10.1142/S0218216520430087 \|issn=0218-2165}}</ref> using highly flexible geometric-topology ~~method~~, far more than having been previously constructed.		In 2020, new and much more complicated Brunnian links have been discovered in <ref name=":1">{{Cite journal \|last=Bai \|first=Sheng \|last2=Wang \|first2=Weibiao \|date=November 2020 \|title=New criteria and constructions of Brunnian links \|url=https://www.worldscientific.com/doi/abs/10.1142/S0218216520430087 \|journal=Journal of Knot Theory and Its Ramifications \|volume=29 \|issue=13 \|pages=2043008 \|doi=10.1142/S0218216520430087 \|issn=0218-2165}}</ref> using highly flexible geometric-topology methods, far more than having been previously constructed. See Section 6 <ref name=":1" />.

	==Non-circularity==		==Non-circularity==
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	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 [[graded Lie algebra]] of the [[lower central series]] of the free group, which can be interpreted as "relations" in the [[free Lie algebra]].		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 [[graded Lie algebra]] of the [[lower central series]] of the free group, which can be interpreted as "relations" in the [[free Lie algebra]].

	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 ~~almostly~~ every Brunnian link.<ref name=":0" /> A geometric classification theorem for Brunnian links was given.<ref name=":0" /> 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.<ref name=":0">{{Cite journal \|last=Bai \|first=Sheng \|last2=Ma \|first2=Jiming \|date=September 2021 \|title=Satellite constructions and geometric classification of Brunnian links \|url=https://www.worldscientific.com/doi/10.1142/S0218216521400058 \|journal=Journal of Knot Theory and Its Ramifications \|volume=30 \|issue=10 \|pages=2140005 \|doi=10.1142/S0218216521400058 \|issn=0218-2165}}</ref>		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.<ref name=":0" /> A geometric classification theorem for Brunnian links was given.<ref name=":0" /> 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.<ref name=":0">{{Cite journal \|last=Bai \|first=Sheng \|last2=Ma \|first2=Jiming \|date=September 2021 \|title=Satellite constructions and geometric classification of Brunnian links \|url=https://www.worldscientific.com/doi/10.1142/S0218216521400058 \|journal=Journal of Knot Theory and Its Ramifications \|volume=30 \|issue=10 \|pages=2140005 \|doi=10.1142/S0218216521400058 \|issn=0218-2165}}</ref>

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BattyBot: Fixed reference date error(s) (see CS1 errors: dates for details) and AWB general fixes

2023-05-29T19:32:08Z

Fixed reference date error(s) (see CS1 errors: dates for details) and AWB general fixes

← Previous revision		Revision as of 19:32, 29 May 2023
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	An example of a ''n''-component Brunnian link is given by the "rubberband" Brunnian Links, where each component is looped around the next as ''aba''<sup>−1</sup>''b''<sup>−1</sup>, with the last looping around the first, forming a circle.<ref>[http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links "Rubberband" Brunnian Links]</ref>		An example of a ''n''-component Brunnian link is given by the "rubberband" Brunnian Links, where each component is looped around the next as ''aba''<sup>−1</sup>''b''<sup>−1</sup>, with the last looping around the first, forming a circle.<ref>[http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links "Rubberband" Brunnian Links]</ref>

	In 2020, new and much more complicated Brunnian links have been discovered in <ref>{{Cite journal \|last=Bai \|first=Sheng \|last2=Wang \|first2=Weibiao \|date=2020~~-11~~ \|title=New criteria and constructions of Brunnian links \|url=https://www.worldscientific.com/doi/abs/10.1142/S0218216520430087 \|journal=Journal of Knot Theory and Its Ramifications \|volume=29 \|issue=13 \|pages=2043008 \|doi=10.1142/S0218216520430087 \|issn=0218-2165}}</ref> using highly flexible geometric-topology method, far more than having been previously constructed.		In 2020, new and much more complicated Brunnian links have been discovered in <ref>{{Cite journal \|last=Bai \|first=Sheng \|last2=Wang \|first2=Weibiao \|date=November 2020 \|title=New criteria and constructions of Brunnian links \|url=https://www.worldscientific.com/doi/abs/10.1142/S0218216520430087 \|journal=Journal of Knot Theory and Its Ramifications \|volume=29 \|issue=13 \|pages=2043008 \|doi=10.1142/S0218216520430087 \|issn=0218-2165}}</ref> using highly flexible geometric-topology method, far more than having been previously constructed.

	==Non-circularity==		==Non-circularity==
Line 41:		Line 41:
	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 [[graded Lie algebra]] of the [[lower central series]] of the free group, which can be interpreted as "relations" in the [[free Lie algebra]].		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 [[graded Lie algebra]] of the [[lower central series]] of the free group, which can be interpreted as "relations" in the [[free Lie algebra]].

	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 almostly every Brunnian link. <ref name=":0" />A geometric classification theorem for Brunnian links was given.<ref name=":0" /> 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.<ref name=":0">{{Cite journal \|last=Bai \|first=Sheng \|last2=Ma \|first2=Jiming \|date=2021~~-09~~ \|title=Satellite constructions and geometric classification of Brunnian links \|url=https://www.worldscientific.com/doi/10.1142/S0218216521400058 \|journal=Journal of Knot Theory and Its Ramifications \|volume=30 \|issue=10 \|pages=2140005 \|doi=10.1142/S0218216521400058 \|issn=0218-2165}}</ref>		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 almostly every Brunnian link.<ref name=":0" /> A geometric classification theorem for Brunnian links was given.<ref name=":0" /> 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.<ref name=":0">{{Cite journal \|last=Bai \|first=Sheng \|last2=Ma \|first2=Jiming \|date=September 2021 \|title=Satellite constructions and geometric classification of Brunnian links \|url=https://www.worldscientific.com/doi/10.1142/S0218216521400058 \|journal=Journal of Knot Theory and Its Ramifications \|volume=30 \|issue=10 \|pages=2140005 \|doi=10.1142/S0218216521400058 \|issn=0218-2165}}</ref>

	=== Massey products ===		=== Massey products ===

133.3.201.132: /* Examples */

2023-05-28T12:30:12Z

Examples

← Previous revision		Revision as of 12:30, 28 May 2023
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	An example of a ''n''-component Brunnian link is given by the "rubberband" Brunnian Links, where each component is looped around the next as ''aba''<sup>−1</sup>''b''<sup>−1</sup>, with the last looping around the first, forming a circle.<ref>[http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links "Rubberband" Brunnian Links]</ref>		An example of a ''n''-component Brunnian link is given by the "rubberband" Brunnian Links, where each component is looped around the next as ''aba''<sup>−1</sup>''b''<sup>−1</sup>, with the last looping around the first, forming a circle.<ref>[http://katlas.math.toronto.edu/wiki/%22Rubberband%22_Brunnian_Links "Rubberband" Brunnian Links]</ref>

	In 2020, new and much more complicated Brunnian links have been discovered in <ref>{{Cite journal \|last=Bai \|first=Sheng \|last2=Wang \|first2=Weibiao \|date=2020-11 \|title=New criteria and constructions of Brunnian links \|url=https://www.worldscientific.com/doi/abs/10.1142/S0218216520430087 \|journal=Journal of Knot Theory and Its Ramifications \|volume=29 \|issue=13 \|pages=2043008 \|doi=10.1142/S0218216520430087 \|issn=0218-2165}}</ref>, far more than having been previously constructed.		In 2020, new and much more complicated Brunnian links have been discovered in <ref>{{Cite journal \|last=Bai \|first=Sheng \|last2=Wang \|first2=Weibiao \|date=2020-11 \|title=New criteria and constructions of Brunnian links \|url=https://www.worldscientific.com/doi/abs/10.1142/S0218216520430087 \|journal=Journal of Knot Theory and Its Ramifications \|volume=29 \|issue=13 \|pages=2043008 \|doi=10.1142/S0218216520430087 \|issn=0218-2165}}</ref> using highly flexible geometric-topology method, far more than having been previously constructed.

	==Non-circularity==		==Non-circularity==

133.3.201.132: /* Classification */

2023-05-28T12:25:53Z

Classification

← Previous revision		Revision as of 12:25, 28 May 2023
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	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 [[graded Lie algebra]] of the [[lower central series]] of the free group, which can be interpreted as "relations" in the [[free Lie algebra]].		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 [[graded Lie algebra]] of the [[lower central series]] of the free group, which can be interpreted as "relations" in the [[free Lie algebra]].

	In 2021, two special satellite operations ~~are~~ investigated, called satellite-sum and satellite-tie, both of which can construct infinitely many distinct Brunnian links from ~~almost~~ every Brunnian link. A geometric classification theorem for Brunnian links is given. More interestingly, ~~and~~ a canonical geometric decomposition in terms of satellite-sum and satellite-tie, which is simpler than JSJ-decomposition, for Brunnian links. The building blocks of Brunnian links ~~then~~ 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.<ref>{{Cite journal \|last=Bai \|first=Sheng \|last2=Ma \|first2=Jiming \|date=2021-09 \|title=Satellite constructions and geometric classification of Brunnian links \|url=https://www.worldscientific.com/doi/10.1142/S0218216521400058 \|journal=Journal of Knot Theory and Its Ramifications \|volume=30 \|issue=10 \|pages=2140005 \|doi=10.1142/S0218216521400058 \|issn=0218-2165}}</ref>		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 almostly every Brunnian link. <ref name=":0" />A geometric classification theorem for Brunnian links was given.<ref name=":0" /> 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.<ref name=":0">{{Cite journal \|last=Bai \|first=Sheng \|last2=Ma \|first2=Jiming \|date=2021-09 \|title=Satellite constructions and geometric classification of Brunnian links \|url=https://www.worldscientific.com/doi/10.1142/S0218216521400058 \|journal=Journal of Knot Theory and Its Ramifications \|volume=30 \|issue=10 \|pages=2140005 \|doi=10.1142/S0218216521400058 \|issn=0218-2165}}</ref>

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