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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 In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...
, an unlink is a link that is equivalent (under
ambient isotopy In the mathematical subject of topology, an ambient isotopy, also called an ''h-isotopy'', is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, on ...
) 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 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 bicond ...
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 In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Ph.D. thesis, . Notably, the link group is not in general the fundamental group of the link comple ...
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 In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf. Geometric realization A concrete model consists of ...
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 ...
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 In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component. As a ...
of ''n'' components such that any proper sublink is an unlink (a Brunnian link). The Whitehead link 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 ...
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 ...


References


Further reading

*Kawauchi, A. ''A Survey of Knot Theory''. Birkhauser. {{Knot theory, state=collapsed