In mathematical knot theory, a link is a collection of knots 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 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 2 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.
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 S n {\displaystyle S^{n}} diffeomorphic to a disjoint union of a finite number of spheres, S j {\displaystyle 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 : X → R 2 × I {\displaystyle T\colon X\to \mathbf {R} ^{2}\times I}
of a (smooth) compact 1-manifold with boundary ( X , ∂ X ) {\displaystyle (X,\partial X)} into the plane times the interval I = [ 0 , 1 ] , {\displaystyle I=[0,1],} such that the boundary T ( ∂ X ) {\displaystyle T(\partial X)} is embedded in
R × { 0 , 1 } {\displaystyle \mathbf {R} \times \{0,1\}} ( { 0 , 1 } = ∂ I {\displaystyle \{0,1\}=\partial I} ). The type of a tangle is the manifold X, together with a fixed embedding of ∂ X . {\displaystyle \partial X.}
Concretely, a connected compact 1-manifold with boundary is an interval I = [ 0 , 1 ] {\displaystyle I=[0,1]} or a circle S 1 {\displaystyle S^{1}} (compactness rules out the open interval ( 0 , 1 ) {\displaystyle (0,1)} and the half-open interval [ 0 , 1 ) , {\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 of n intervals I = [ 0 , 1 ] {\displaystyle I=[0,1]} and m circles S 1 . {\displaystyle S^{1}.} The condition that the boundary of X lies in
R × { 0 , 1 } {\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
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