  A knot is an an embedding of a circle into three-dimensional space. We say
that a knot is unknotted if there is an ambient isotopy of the embedding to a
standard circle. By representing knots via planar diagrams, we discuss the
problem of unknotting a knot diagram when we know that it is unknotted. This
problem is surprisingly difficult, since it has been shown that knot diagrams
may need to be made more complicated before they may be simplified. We do not
yet know, however, how much more complicated they must get. We give an
introduction to the work of Dynnikov who discovered the key use of
arc--presentations to solve the problem of finding a way to detect the unknot
directly from a diagram of the knot. Using Dynnikov's work, we show how to
obtain a quadratic upper bound for the number of crossings that must be
introduced into a sequence of unknotting moves. We also apply Dynnikov's
results to find an upper bound for the number of moves required in an
unknotting sequence.
