Unknot - Unknotting Problem

Unknotting Problem

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 (not using invariants), 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.