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Knot theory

A three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot
A knot diagram of the trefoil knot

In [mathematics] , knot theory is the area of [topology] that studies [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 to prevent it from becoming undone. In precise mathematical language, a knot is an [embedding] of a [circle] in 3-dimensional [Euclidean space] , R 3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R 3 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.

Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram. Any given knot can be drawn in many different ways using a knot diagram. 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 [complexity] . In practice, knots are often distinguished by 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.

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