Unlink

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.

Properties

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

References

• Kawauchi, A. A Survey of knot theory. Birkhauser.

See also

• Unknot
• Link (knot theory)
• Linking number