In the mathematical field of knot theory, the unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane.

  • 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 isotopic to the unknot.
  • 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.
  • Kanenobu has shown that for all n > 1 there exists a hyperbolic link of n components such that any proper sublink is an unlink. The Whitehead link and Borromean rings are such examples for n = 2, 3.

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