in the number of crossings. Therefore, a brute force search for all sequences of Reidemeister moves can detect unknottedness in exponential time. Similarly, any two triangulations of the same knot complement may be connected by a sequence of Pachner moves of length at most doubly exponential in the number of crossings. Therefore, it is possible to determine whether a knot is the unknot by testing all sequences of Pachner moves of this length, starting from the complement of the given knot, and determining whether any of them transforms the complement into a standard triangulation of a solid torus. The time for this method would be triply exponential; however, experimental evidence suggests that this bound is very pessimistic and that many fewer Pachner moves are needed. Any arc-presentation of an unknot can be monotonically simplified to a minimal one using elementary moves. So a brute force search among all arc-presentations of not greater complexity gives a single-exponential algorithm for the unknotting problem. Residual finiteness of the knot group (which follows from geometrization of Haken manifolds) gives an algorithm: check if the group has non-cyclic finite group quotient. This idea is used in Kuperberg's result that the unknotting problem is in co-NP. Knot Floer homology of the knot detects the genus of the knot, which is 0 if and only if the knot is an unknot. A combinatorial version of knot Floer homology allows it to be computed . Khovanov homology detects the unknot according to a result of Kronheimer and Mrowka. The complexity of Khovanov homology at least as high as the #P-hard problem of computing the Jones polynomial, but it may be calculated in practice using an algorithm and program of . Bar-Natan provides no rigorous analysis of his algorithm, but heuristically estimates it to be exponential in the pathwidth of a crossing diagram, which in turn is at most proportional to the square root of the number of crossings. Understanding the complexity of these algorithms is an active field of study. See also Algorithmic topology Unknotting number Notes References . . . . . . . . . . . . . . Knot theory 3-manifolds Geometric topology Computational problems Unsolved problems in geometry