Knot theory is a branch of topology inspired by observations, as the name suggests, of common knots. But progress in the field does not depend exclusively on experiments with twine. Knot theory concerns itself with abstract properties of theoretical knots — the spatial arrangements that in principle could be assumed by a loop of string.

When mathematical topologists consider knots and other entanglements such as links and braids, they describe how the knot is positioned in the space around it, called the ambient space. If the knot is moved smoothly to a different position in the ambient space, then the knot is considered to be unchanged, and if one knot can be moved smoothly to coincide with another knot, the two knots are called "equivalent".

In mathematical language, knots are embeddings of the circle in three-dimensional space. A mathematical knot thus resembles an ordinary knot with its ends spliced. The topological theory of knots investigates such questions as whether two knots can be smoothly moved to match one another, without opening the splice. The question of untying an ordinary knot has to do with unwedging tangles of rope pulled tight, but this concept plays at best a minor role in the mathematical theory. A knot can be untied in the topological sense if and only if it can be smoothly moved through the ambient space until it assumes the shape of a circle. If this can be done, the knot is called the unknot.

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