  For an oriented link diagram D, the warping degree d(D) is the smallest
number of crossing changes which are needed to obtain a monotone diagram from
D. We show that d(D)+d(-D)+sr(D) is less than or equal to the crossing number
of D, where -D denotes the inverse of D and sr(D) denotes the number of
components which have at least one self-crossing. Moreover, we give a necessary
and sufficient condition for the equality. We also consider the minimal
d(D)+d(-D)+sr(D) for all diagrams D. For the warping degree and linking warping
degree, we show some relations to the linking number, unknotting number, and
the splitting number.
