Unlink
Encyclopedia
In the mathematical
field of knot theory
, the unlink is a link
that is equivalent (under ambient isotopy
) to finitely many disjoint circles in the plane.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, the unlink is a link
Link (knot theory)
In mathematics, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory...
that is equivalent (under ambient isotopy
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", a manifold, taking a submanifold to another submanifold. For example in knot theory, one considers two knots the same if one can distort one knot into the...
) 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 ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is the unknotUnknotThe unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the...
. - The link groupLink groupIn 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 Bachelor's thesis, .- Definition :...
of an n-component unlink is the free groupFree groupIn mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
on n generators, and is used in classifying Brunnian links.
Examples
- The Hopf linkHopf linkthumb|right|[[Skein relation]] for the Hopf link.In mathematical knot theory, the Hopf link, named after Heinz Hopf, is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once...
is a simple example of a link with two components that is not an unlink. - The Borromean ringsBorromean ringsIn mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link, i.e., removing any ring results in two unlinked rings.- Mathematical properties :...
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. - Kanenobu has shown that for all n > 1 there exists a hyperbolic linkHyperbolic linkIn 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...
of n components such that any proper sublink is an unlink (a Brunnian linkBrunnian linkIn knot theory, a branch of mathematics, a Brunnian link is a nontrivial link that becomes trivial if any component is removed. In other words, cutting any loop frees all the other loops ....
). The Whitehead linkWhitehead linkIn knot theory, the Whitehead link, discovered by J.H.C. Whitehead, is one of the most basic links.J.H.C. Whitehead spent much of the 1930s looking for a proof of the Poincaré conjecture...
and Borromean ringsBorromean ringsIn mathematics, the Borromean rings consist of three topological circles which are linked and form a Brunnian link, i.e., removing any ring results in two unlinked rings.- Mathematical properties :...
are such examples for n = 2, 3.