Editing Geometric topology (section)

===Knot theory===
{{main|Knot theory}}
[[Knot theory]] is the study of [[knot (mathematics)|mathematical knot]]s. While inspired by knots which 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.

To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other [[3-manifold|three-dimensional spaces]] and objects other than circles can be used; see ''[[knot (mathematics)]]''. Higher-dimensional knots are [[n-sphere|''n''-dimensional spheres]] in ''m''-dimensional Euclidean space.