  Deciding whether a diagram of a knot can be untangled with a given number of
moves (as a part of the input) is known to be NP-complete. In this paper we
determine the parameterized complexity of this problem with respect to a
natural parameter called defect. Roughly speaking, it measures the efficiency
of the moves used in the shortest untangling sequence of Reidemeister moves.
  We show that the II- moves in a shortest untangling sequence can be
essentially performed greedily. Using that, we show that this problem belongs
to W[P] when parameterized by the defect. We also show that this problem is
W[P]-hard by a reduction from Minimum axiom set.
