Unlink

This article is about the mathematical concept. For the Unix system call, see unlink (Unix).
Unlink
2-component unlink
Common nameCircle
Crossing no.0
Linking no.0
Stick no.6
Unknotting no.0
Conway notation-
A–B notation02
1
Dowker notation-
NextL2a1
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

  • 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

  • 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

References

  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 0833204

Further reading

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