# Unlink

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> Markdown URL: https://mediated.wiki/source/Unlink.md
> Source: https://en.wikipedia.org/wiki/Unlink
> Source revision: 1215582099
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{{short description|Link that consists of finitely many unlinked unknots}}
{{about|the mathematical concept|the Unix system call|unlink (Unix)}}
{{Infobox knot theory
| name=              Unlink
| practical name=    Circle
| image=             Unlink.png
| caption=           2-component unlink
| arf invariant=     
| bridge number=     
| crossing number=   0
| linking number=    0
| stick number=      6
| unknotting number= 0
| conway_notation=   -
| ab_notation=       0{{sup sub|2|1}}
| dowker notation=   -
| thistlethwaite=    
| other=             
 | alternating=      
 | amphichiral=      
 | fibered=          
 | slice=            
 | tricolorable=     tricolorable (if n>1)
| last link=         
| next link=         L2a1
}}
{{Wiktionary}}

In the [mathematical](/source/mathematics) field of [knot theory](/source/knot_theory), an '''unlink''' is a [link](/source/Link_(knot_theory)) that is equivalent (under [ambient isotopy](/source/ambient_isotopy)) to finitely many disjoint circles in the plane.<ref name=":0" />

The '''two-component unlink''', consisting of two non-interlinked [unknots](/source/Unknot), is the simplest possible unlink.

== Properties ==
* An ''n''-component link ''L''&nbsp;⊂&nbsp;'''S'''<sup>3</sup> is an unlink if and only if there exists ''n'' disjointly embedded discs ''D''<sub>''i''</sub>&nbsp;⊂&nbsp;'''S'''<sup>3</sup> such that ''L''&nbsp;=&nbsp;∪<sub>''i''</sub>∂''D''<sub>''i''</sub>.
* A link with one component is an unlink [if and only if](/source/if_and_only_if) it is the [unknot](/source/unknot).
* The [link group](/source/link_group) of an ''n''-component unlink is the [free group](/source/free_group) on ''n'' generators, and is used in classifying [Brunnian links](/source/Brunnian_links).

== Examples ==
* The [Hopf link](/source/Hopf_link) is a simple example of a link with two components that is not an unlink.
* The [Borromean rings](/source/Borromean_rings) form a link with three components that is not an unlink; however, any two of the rings considered on their own do form a two-component unlink.
* Taizo Kanenobu has shown that for all ''n''&nbsp;&gt;&nbsp;1 there exists a [hyperbolic link](/source/hyperbolic_link) of ''n'' components such that any proper sublink is an unlink (a [Brunnian link](/source/Brunnian_link)). The [Whitehead link](/source/Whitehead_link) and [Borromean rings](/source/Borromean_rings) are such examples for ''n''&nbsp;=&nbsp;2, 3.<ref name=":0">{{citation
 | last = Kanenobu | first = Taizo
 | doi = 10.2969/jmsj/03820295
 | issue = 2
 | journal = Journal of the Mathematical Society of Japan
 | mr = 833204
 | pages = 295–308
 | title = Hyperbolic links with Brunnian properties
 | volume = 38
 | year = 1986| doi-access = free
 }}</ref>

==See also==
*[Linking number](/source/Linking_number)

==References==
{{reflist}}

==Further reading==
*Kawauchi, A. ''A Survey of Knot Theory''. Birkhauser.

{{Knot theory|state=collapsed}}

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Adapted from the Wikipedia article [Unlink](https://en.wikipedia.org/wiki/Unlink) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Unlink?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.