Unlink

Link that consists of finitely many unlinked unknots

This article is about the mathematical concept. For the Unix system call, see unlink (Unix).

Unlink

2-component unlink

Common name| Circle
Crossing no.| 0
Linking no.| 0
Stick no.| 6
Unknotting no.| 0
Conway notation| -
A–B notation| 02

Dowker notation| -
Next| L2a1
Other
, tricolorable (if n>1)

Look up unlink in Wiktionary, the free dictionary.

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.[1]

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

## Properties

[edit]

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

[edit]

* 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.[1]

## See also

[edit]

* Linking number

## References

[edit]

1. ^ a b Kanenobu, Taizo (1986), "Hyperbolic links with Brunnian properties", Journal of the Mathematical Society of Japan, 38 (2): 295–308, doi:10.2969/jmsj/03820295, MR 0833204295-308&rft.date=1986&rft_id=info:doi/10.2969/jmsj/03820295&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=833204#id-name=MR&rft.aulast=Kanenobu&rft.aufirst=Taizo&rfr_id=info:sid/en.wikipedia.org:Unlink" class="Z3988">

## Further reading

[edit]

* Kawauchi, A. A Survey of Knot Theory. Birkhauser.

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