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Selected Article

A 3-D depiction of a trefoil knot

Knot theory is the mathematical branch of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional Euclidean space, R³. This is basically equivalent to a conventional knotted string with the ends joined together to prevent it from becoming undone. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R³ upon itself. 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. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot. One way of distinguishing knots is by using a knot invariant, a "quantity" which remains the same even with different descriptions of a knot.