In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
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'' ⊂ S
3 is an unlink if and only if there exists ''n'' disjointly embedded discs ''D''
''i'' ⊂ S
3 such that ''L'' = ∪
''i''∂''D''
''i''.
* A link with one component is an unlink
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
it is the
unknot
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...
.
* The
link group of an ''n''-component unlink is the
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
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
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the t ...
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
In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from the overlay of a circle and a figure-eight shaped loop.
Structure
A common w ...
and
Borromean rings
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the t ...
are such examples for ''n'' = 2, 3.
See also
*
Linking number
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In E ...
References
Further reading
*Kawauchi, A. ''A Survey of Knot Theory''. Birkhauser.
{{Knot theory, state=collapsed