High Quality Content by WIKIPEDIA articles! The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, ambient-isotoped, to the standard unknot, the embedding of the circle as a geometrically round circle. The unknot is also called the trivial knot. An unknot is the identity element with respect to the knot sum operation. A tricky unknot diagram by Morwen Thistlethwaite Deciding if a particular knot is the unknot was a major driving force behind knot invariants, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram. Currently there are several well-known unknot recognition algorithms, but they are either known to be inefficient or have no efficient implementation. It is not known whether many of the current invariants, such as finite type invariants, are a complete invariant of the unknot, but knot Floer homology is known to detect the unknot. Even if they were, the problem of computing them efficiently remains.

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