Unlink

Unlink
2-component unlink
Common name Circle
Crossing no. 0
Linking no. 0
Stick no. 6
Unknotting no. 0
Conway notation -
A-B notation 02
1
Dowker notation -
Next L2a1
Other
, tricolorable (if n>1)

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 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.
  • 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.

See also

Further reading

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