Unlink

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.

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)

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]
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See also

References
  1. 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.
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