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Unlink

Link that consists of finitely many unlinked unknots From Wikipedia, the free encyclopedia

Unlink
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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]

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

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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 = iDi.
  • 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

Further reading

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