Knot theory
Knotting spheres of higher dimension

Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a two-dimensional sphere (S 2 {\displaystyle \mathbb {S} ^{2}}) embedded in 4-dimensional Euclidean space (R 4 {\displaystyle \mathbb {R} ^{4}}). Such an embedding is knotted if there is no homeomorphism of R 4 {\displaystyle \mathbb {R} ^{4}} onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere. Suspended knots and spun knots are two typical families of such 2-sphere knots.

The mathematical technique called "general position" implies that for a given n -sphere in m -dimensional Euclidean space, if m is large enough (depending on n), the sphere should be unknotted. In general, piecewise-linear n -spheres form knots only in (n + 2)-dimensional space (Zeeman 1963), although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted (4 k − 1) {\displaystyle (4k-1)} -spheres in 6 k -dimensional space; e.g., there is a smoothly knotted 3-sphere in R 6 {\displaystyle \mathbb {R} ^{6}} (Haefliger 1962) (Levine 1965). Thus the codimension of a smooth knot can be arbitrarily large when not fixing the dimension of the knotted sphere; however, any smooth k -sphere embedded in R n {\displaystyle \mathbb {R} ^{n}} with 2 n − 3 k − 3 > 0 {\displaystyle 2n-3k-3>0} is unknotted. The notion of a knot has further generalisations in mathematics, see: Knot (mathematics), isotopy classification of embeddings.

Every knot in the n -sphere S n {\displaystyle \mathbb {S} ^{n}} is the link of a real-algebraic set with isolated singularity in R n + 1 {\displaystyle \mathbb {R} ^{n+1}} (Akbulut & King 1981).

An n -knot is a single S n {\displaystyle \mathbb {S} ^{n}} embedded in R m {\displaystyle \mathbb {R} ^{m}}. An n -link consists of k -copies of S n {\displaystyle \mathbb {S} ^{n}} embedded in R m {\displaystyle \mathbb {R} ^{m}}, where k is a natural number. Both the m = n + 2 {\displaystyle m=n+2} and the m > n + 2 {\displaystyle m>n+2} cases are well studied, and so is the n > 1 {\displaystyle n>1} case.
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