{{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 field of knot theory, an '''unlink''' is a 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 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}}