| 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, an 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.
- 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]
See also
References
- ^ Kanenobu, Taizo (1986), "Hyperbolic links with Brunnian properties", Journal of the Mathematical Society of Japan, 38 (2): 295–308, doi:10.2969/jmsj/03820295, MR 0833204
Further reading
- Kawauchi, A. A Survey of Knot Theory. Birkhauser.