Editing Knot theory (section)

{{short description|Study of mathematical knots}}
[[File:Tabela de nós matemáticos 01, crop.jpg|thumb|Examples of different knots including the [[trivial knot]] (top left) and the [[trefoil knot]] (below it)]]
[[File:TrefoilKnot 01.svg|thumb|A knot diagram of the trefoil knot, the simplest non-trivial knot]]

In [[topology]], '''knot theory''' is the study of [[knot (mathematics)|mathematical knot]]s. While inspired by [[knot]]s which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring (or "[[unknot]]"). In mathematical language, a knot is an [[embedding]] of a [[circle]] in 3-dimensional [[Euclidean space]], <math>\mathbb{E}^3</math>. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of <math>\mathbb{R}^3</math> upon itself (known as an [[ambient isotopy]]); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.

Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.

A complete algorithmic solution to this problem exists, which has unknown [[Computational complexity|complexity]].<ref>As first sketched using the theory of [[Haken manifold]]s by {{harvtxt|Haken|1962}}. For a more recent survey, see {{harvtxt|Hass|1998}}</ref> In practice, knots are often distinguished using a ''[[knot invariant]]'', a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include [[knot polynomials]], [[knot group]]s, and hyperbolic invariants.

The original motivation for the founders of knot theory was to create a table of knots and [[link (knot theory)|link]]s, which are knots of several components entangled with each other. More than six billion knots and links [[knot tabulation|have been tabulated]] since the beginnings of knot theory in the 19th century.

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)]]''. For example, a higher-dimensional knot is an [[n-sphere|''n''-dimensional sphere]] embedded in (''n''+2)-dimensional Euclidean space.