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

}}

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.

The two-component unlink, consisting of two non-interlinked unknots, is the simplest possible unlink.

Properties

An n-component link L ⊂ S³ is an unlink if and only if there exists n disjointly embedded discs D_i ⊂ S³ such that L = ∪_i∂D_i.
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.

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 > 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 = 2, 3.{{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

}}