In
mathematical 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 ...
, a link is a collection of
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ...
s which do not intersect, but which may be linked (or knotted) 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
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 ...
. Implicit in this definition is that there is a ''trivial'' reference link, usually called the
unlink, but the word is also sometimes used in context where there is no notion of a trivial link.
For example, a
co-dimension 2 link in 3-dimensional space is a
subspace of 3-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
(or often the
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
) whose
connected components are
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
s.
The simplest nontrivial example of a link with more than one component is called 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 o ...
, which consists of two circles (or
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 ...
s) linked together once. The circles in
the
Borromean rings are collectively linked despite the fact that no two of them are directly linked. The Borromean rings thus form a
Brunnian link and in fact constitute the simplest such link.
Generalizations
The notion of a link can be generalized in a number of ways.
General manifolds
Frequently the word link is used to describe any submanifold of the
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
diffeomorphic to a disjoint union of a finite number of
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
s,
.
In full generality, the word link is essentially the same as the word ''knot'' – the context is that one has a submanifold ''M'' of a manifold ''N'' (considered to be trivially embedded) and a non-trivial embedding of ''M'' in ''N'', non-trivial in the sense that the 2nd embedding is not
isotopic to the 1st. If ''M'' is disconnected, the embedding is called a link (or said to be linked). If ''M'' is connected, it is called a knot.
Tangles, string links, and braids
While (1-dimensional) links are defined as embeddings of circles, it is often interesting and especially technically useful to consider embedded intervals (strands), as in
braid theory.
Most generally, one can consider a tangle
– a tangle is an embedding
:
of a (smooth) compact 1-manifold with boundary
into the plane times the interval
such that the boundary
is embedded in
:
(
).
The type of a tangle is the manifold ''X,'' together with a fixed embedding of
Concretely, a connected compact 1-manifold with boundary is an interval