Editing Knot theory (section)

===Knotting spheres of higher dimension===
Since a knot can be considered topologically a 1-dimensional sphere, the next generalization is to consider a [[2-sphere|two-dimensional sphere]] (<math>\mathbb{S}^2</math>) embedded in 4-dimensional Euclidean space (<math>\R^4</math>). Such an embedding is knotted if there is no homeomorphism of <math>\R^4</math> onto itself taking the embedded 2-sphere to the standard "round" embedding of the 2-sphere.  [[Suspended knot]]s and [[spun knot]]s 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 manifold|piecewise-linear]] [[n-sphere|''n''-sphere]]s form knots only in (''n''&nbsp;+&nbsp;2)-dimensional space {{Harv|Zeeman|1963}}, although this is no longer a requirement for smoothly knotted spheres. In fact, there are smoothly knotted <math>(4k-1)</math>-spheres in 6''k''-dimensional space; e.g.,  there is a smoothly knotted 3-sphere in <math>\R^6</math> {{Harv|Haefliger|1962}} {{Harv|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 <math>\R^n</math> with <math>2n-3k-3>0</math> is unknotted. The notion of a knot has further generalisations in mathematics, see: [[Knot (mathematics)]], [[Whitney embedding theorem#Isotopy versions|isotopy classification of embeddings]].

Every knot in the [[n-sphere|''n''-sphere]] <math>\mathbb{S}^n</math> is the link of a [[real algebraic set|real-algebraic set]] with isolated singularity in <math>\R^{n+1}</math> {{Harv|Akbulut|King|1981}}.

An ''n''-knot is a single <math>\mathbb{S}^n</math> embedded in  <math>\R^m</math>. An ''n''-link consists of ''k''-copies of <math>\mathbb{S}^n</math> embedded in <math>\R^m</math>, where ''k'' is a [[natural number]]. Both the <math>m=n+2</math> and the <math>m>n+2</math> cases are well studied, and so is the <math>n>1</math> case.<ref>{{citation |first1=J. |last1=Levine |first2=K |last2=Orr |chapter=A survey of applications of surgery to knot and link theory |citeseerx = 10.1.1.64.4359 |title=Surveys on Surgery Theory: Papers Dedicated to C.T.C. Wall |volume=1 |series=Annals of mathematics studies |publisher=[[Princeton University Press]] |year=2000 |isbn=978-0691049380 }} — An introductory article to high dimensional knots and links for the advanced readers</ref><ref>{{citation |first=Eiji |last=Ogasa |arxiv=1304.6053 |title=Introduction to high dimensional knots |bibcode=2013arXiv1304.6053O |year=2013 }} — An introductory article to high dimensional knots and links for beginners</ref>