In mathematical knot theory, 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. 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.

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

Frequently the word **link** is used to describe any submanifold of the sphere diffeomorphic to a disjoint union of a finite number of spheres, .

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.

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**^{ [1] }^{ [2] } – 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 or a circle (compactness rules out the open interval and the half-open interval 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 and *m* circles The condition that the boundary of *X* lies in

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

- ( and ),

and then being able to move in a two-dimensional horizontal direction ()

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, 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, 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, for instance. Compare with closed braids.

In 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 together so that it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

In topology, a branch of mathematics, two continuous functions from one topological space to another are called **homotopic** if one can be "continuously deformed" into the other, such a deformation being called a **homotopy** between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

In mathematics, **Thurston's geometrization conjecture** states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

In mathematics, a **knot** is an embedding of a topological circle *S*^{1} in 3-dimensional Euclidean space, **R**^{3}, considered up to continuous deformations (isotopies).

In mathematics, the **linking number** is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. The linking number is always an integer, but may be positive or negative depending on the orientation of the two curves.

In mathematics, the **braid group on n strands**, also known as the

In mathematics, **geometric topology** is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

In mathematics, **low-dimensional topology** is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

In mathematics, in the subfield of geometric topology, the **mapping class group** is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.

In mathematics, a **3-manifold** is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

In the mathematical field of knot theory, the **Jones polynomial** is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable with integer coefficients.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

In mathematics, specifically in geometric topology, **surgery theory** is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor (1961). Originally developed for differentiable manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds.

In surgery theory, a branch of mathematics, the **stable normal bundle** of a differentiable manifold is an invariant which encodes the stable normal data. There are analogs for generalizations of manifold, notably PL-manifolds and topological manifolds. There is also an analogue in homotopy theory for Poincaré spaces, the **Spivak spherical fibration**, named after Michael Spivak.

In mathematics, two links and are **concordant** if there exists an embedding such that and .

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,.

In the mathematical field of low-dimensional topology, a **clasper** is a surface in a 3-manifold on which surgery can be performed.

In mathematics, a **tangle** is generally one of two related concepts:

In mathematics, a **configuration space** is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space. In mathematics, they are used to describe assignments of a collection of points to positions in a topological space. More specifically, configuration spaces in mathematics are particular examples of configuration spaces in physics in the particular case of several non-colliding particles.

- ↑ Habegger, Nathan; Lin, X.S. (1990), "The classification of links up to homotopy",
*Journal of the American Mathematical Society*, 2, American Mathematical Society,**3**(2): 389–419, doi: 10.2307/1990959 , JSTOR 1990959 - ↑ Habegger, Nathan; Masbaum, Gregor (2000), "The Kontsevich integral and Milnor's invariants",
*Topology*,**39**(6): 1253–1289, doi: 10.1016/S0040-9383(99)00041-5 , preprint.

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