Definitions

In mathematics, a link is a collection of knots 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. 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 two link in 3-dimensional space is a subspace of 3-dimensional Euclidean space (or often the 3-sphere) whose connected components are homeomorphic to circles.

The simplest nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles (or unknots) linked together once. Borromean rings form a link with three components each equivalent to the unknot. The three loops are collectively linked despite the fact that no two of them are directly linked.

## More generally

Frequently the word link is used to describe any submanifold of the sphere $S^n$ diffeomorphic to a disjoint union of a finite number of spheres, $S^j$.

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.