Editing Low-dimensional topology (section)

===Knot and braid theory===
{{Main|Knot theory|Braid theory}}

[[Knot theory]] is the study of [[knot (mathematics)|mathematical knot]]s. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an [[embedding]] of a [[circle]] in 3-dimensional [[Euclidean space]], '''R'''<sup>3</sup> (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its [[homeomorphism]]s). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of '''R'''<sup>3</sup> upon itself (known as an [[ambient isotopy]]); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

[[Knot complement]]s are frequently-studied 3-manifolds. The knot complement of a [[tame knot]] ''K'' is the three-dimensional space surrounding the knot. To make this precise, suppose that ''K'' is a knot in a three-manifold ''M'' (most often, ''M'' is the  [[3-sphere]]).  Let ''N'' be a [[tubular neighborhood]] of ''K''; so ''N'' is a [[solid torus]].  The knot complement is then the [[complement (set theory)|complement]] of ''N'',
:<math>X_K = M - \mbox{interior}(N).</math>

A related topic is [[braid theory]]. Braid theory is an abstract [[geometry|geometric]] [[theory]] studying the everyday [[braid]] concept, and some generalizations. The idea is that braids can be organized into [[group (mathematics)|group]]s, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second on the twisted strings'. Such groups may be described by explicit [[presentation of a group|presentation]]s, as was shown by {{harvs|first=Emil|last=Artin|authorlink=Emil Artin|year=1947|txt}}.<ref>{{citation
 | last = Artin | first = E. | authorlink = Emil Artin
 | doi = 10.2307/1969218
 | journal = [[Annals of Mathematics]]
 | mr = 0019087
 | pages = 101–126
 | series = Second Series
 | title = Theory of braids
 | volume = 48
 | year = 1947}}.</ref> For an elementary treatment along these lines, see the article on [[braid group]]s. Braid groups may also be given a deeper mathematical interpretation: as the [[fundamental group]] of certain [[Configuration space (mathematics)|configuration space]]s.