View source for Unlink

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

The '''two-component unlink''', consisting of two non-interlinked [[Unknot|unknots]], 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]] it is the [[unknot]].
* The [[link group]] of an ''n''-component unlink is the [[free group]] on ''n'' generators, and is used in classifying [[Brunnian links]].

== Examples ==
* The [[Hopf link]] is a simple example of a link with two components that is not an unlink.
* The [[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]] of ''n'' components such that any proper sublink is an unlink (a [[Brunnian link]]). The [[Whitehead link]] and [[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]]

==References==
{{reflist}}

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

{{Knot theory|state=collapsed}}