In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

knot theory In topology, knot theory is the study of knot (mathematics), 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 be und ...

, 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, o ...

) 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 ''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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

it is the

unknot In the knot theory, 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 (mathematics), knot tied into it, unknotted. To a knot ...

. * 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''− ...

on ''n'' generators, and is used in classifying Brunnian links.

Examples

* The

Hopf link In mathematics, mathematical knot theory, the Hopf link is the simplest nontrivial link (knot theory), link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf. Geometric realizat ...

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 link (knot theory), topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops wh ...

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 c ...

of ''n'' components such that any proper sublink is an unlink (a Brunnian link). The Whitehead link and

are such examples for ''n'' = 2, 3.

Properties

Examples

See also

References

Further reading