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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 ...
S^n 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, 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.


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 :T\colon X \to \mathbf^2 \times I of a (smooth) compact 1-manifold with boundary (X,\partial X) into the plane times the interval I= ,1 such that the boundary T(\partial X) is embedded in :\mathbf \times \ (\ = \partial I). The type of a tangle is the manifold ''X,'' together with a fixed embedding of \partial X. Concretely, a connected compact 1-manifold with boundary is an interval I= ,1/math> or a circle S^1 (compactness rules out the open interval (0,1) and the half-open interval [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 of ''n'' intervals I= ,1/math> and ''m'' circles S^1. The condition that the boundary of ''X'' lies in :\mathbf \times \ 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 :(\mathbf \times 0 and \mathbf \times 1), and then being able to move in a two-dimensional horizontal direction (\mathbf^2) between these lines; one can project these to form a tangle diagram, analogous to a knot diagram. Tangles include links (if ''X'' 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. A string 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 has ''ℓ'' 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 an overhand knot. A braid that is also a string link is called a
pure braid Pure may refer to: Computing * A pure function * A pure virtual function * PureSystems, a family of computer systems introduced by IBM in 2012 * Pure Software, a company founded in 1991 by Reed Hastings to support the Purify tool * Pure-FTPd, ...
, 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 a
tensor category In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and ...
, 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 fixed ''ℓ,'' isotopy classes of ''ℓ''-component string links form a monoid (one can compose all ''ℓ''-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 also ''homotopy'' 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 for
Milnor's invariants In 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 Ph.D. thesis, . Notably, the link group is not in general the fundamental group of the link comple ...
, for instance. Compare with
closed braids Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
.


See also

* Hyperbolic link * Unlink * Link group


References

{{DEFAULTSORT:Link (Knot Theory) Manifolds