Inmathematicalknot theory, alink is a collection ofknots that 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 calledknot theory. Implicit in this definition is that there is atrivial reference link, usually called theunlink, but the word is also sometimes used in context where there is no notion of a trivial link.

For example, aco-dimension 2 link in 3-dimensional space is asubspace of 3-dimensionalEuclidean space (or often the3-sphere) whoseconnected components arehomeomorphic tocircles.

The simplest nontrivial example of a link with more than one component is called theHopf link, which consists of two circles (orunknots) linked together once. The circles intheBorromean rings are collectively linked despite the fact that no two of them are directly linked. The Borromean rings thus form aBrunnian link and in fact constitute the simplest such link.

The notion of a link can be generalized in a number of ways.

Frequently the wordlink is used to describe any submanifold of thesphere${\displaystyle S^n}$ diffeomorphic to a disjoint union of a finite number ofspheres,${\displaystyle S^j}$.

In full generality, the wordlink is essentially the same as the wordknot – the context is that one has a submanifoldM of a manifoldN (considered to be trivially embedded) and a non-trivial embedding ofM inN, non-trivial in the sense that the 2nd embedding is notisotopic to the 1st. IfM is disconnected, the embedding is called a link (or said to belinked). IfM is connected, it is called a knot.

While (1-dimensional) links are defined as embeddings of circles, it is often interesting and especially technically useful to consider embedded intervals (strands), as inbraid theory.

Most generally, one can consider atangle^[1][2] – a tangle is an embedding

${\displaystyle T:X\to {\mathbf{R}}^2\times I}$

of a (smooth) compact 1-manifold with boundary${\displaystyle (X,\mathrm{\partial }X)}$ into the plane times the interval${\displaystyle I=[0,1],}$ such that the boundary${\displaystyle T(\mathrm{\partial }X)}$ is embedded in

${\displaystyle \mathbf{R}\times \{0,1\}}$ (${\displaystyle \{0,1\}=\mathrm{\partial }I}$).

Thetype of a tangle is the manifoldX, together with a fixed embedding of${\displaystyle \mathrm{\partial }X.}$

Concretely, a connected compact 1-manifold with boundary is an interval${\displaystyle I=[0,1]}$ or a circle${\displaystyle S^1}$ (compactness rules out the open interval${\displaystyle (0,1)}$ and the half-open interval${\displaystyle [0,1),}$ neither of which yields non-trivial embeddings since the open end means that they can be shrunk to a point), so a possibly disconnected compact 1-manifold is a collection ofn intervals${\displaystyle I=[0,1]}$ andm circles${\displaystyle S^1.}$ The condition that the boundary ofX lies in

${\displaystyle \mathbf{R}\times \{0,1\}}$

says that intervals either connect two lines or connect two points on one of the lines, but imposes no conditions on the circles.One may view tangles as having a vertical direction (I), lying between and possibly connecting two lines

(${\displaystyle \mathbf{R}\times 0}$ and${\displaystyle \mathbf{R}\times 1}$),

and then being able to move in a two-dimensional horizontal direction (${\displaystyle {\mathbf{R}}^2}$)

between these lines; one can project these to form atangle diagram, analogous to aknot diagram.

Tangles include links (ifX consists of circles only), braids, and others besides – for example, a strand connecting the two lines together with a circle linked around it.

In this context, a braid is defined as a tangle which is always going down – whose derivative always has a non-zero component in the vertical (I) direction. In particular, it must consist solely of intervals, and not double back on itself; however, no specification is made on where on the line the ends lie.

Astring link is a tangle consisting of only intervals, with the ends of each strand required to lie at (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), ... – i.e., connecting the integers, and ending in the same order that they began (one may use any other fixed set of points); if this hasℓ components, we call it an "ℓ-component string link". A string link need not be a braid – it may double back on itself, such as a two-component string link that features anoverhand knot. A braid that is also a string link is called apure braid, and corresponds with the usual such notion.

The key technical value of tangles and string links is that they have algebraic structure. Isotopy classes of tangles form atensor category, where for the category structure, one can compose two tangles if the bottom end of one equals the top end of the other (so the boundaries can be stitched together), by stacking them – they do not literally form a category (pointwise) because there is no identity, since even a trivial tangle takes up vertical space, but up to isotopy they do. The tensor structure is given by juxtaposition of tangles – putting one tangle to the right of the other.

For a fixedℓ, isotopy classes ofℓ-component string links form a monoid (one can compose allℓ-component string links, and there is an identity), but not a group, as isotopy classes of string links need not have inverses. However,concordance classes (and thus alsohomotopy classes) of string links do have inverses, where inverse is given by flipping the string link upside down, and thus form a group.

Every link can be cut apart to form a string link, though this is not unique, and invariants of links can sometimes be understood as invariants of string links – this is the case forMilnor's invariants, for instance. Compare withclosed braids.

• Hyperbolic link
• Unlink
• Link group

• Links (knot theory)
• Manifolds

• Articles with short description
• Short description is different from Wikidata