Unlink Explained
| Unlink |
| Practical Name: | Circle |
| Crossing Number: | 0 |
| Linking Number: | 0 |
| Stick Number: | 6 |
| Unknotting Number: | 0 |
| Conway Notation: | - |
| Ab Notation: | 0 |
| Dowker Notation: | - |
| Tricolorable: | tricolorable (if n>1) |
| Next Link: | L2a1 |
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.
The two-component unlink, consisting of two non-interlinked unknots, is the simplest possible unlink.
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.
See also
Further reading
- Kawauchi, A. A Survey of Knot Theory. Birkhauser.
