  The warping degree of an oriented knot diagram is the minimal number of
crossings which we meet as an under-crossing first when we travel along the
diagram from a fixed point. The warping degree of a knot projection is the
minimal value of the warping degree for all oriented alternating diagrams
obtained from the knot projection. In this paper, we consider the maximal
number of regions which share no crossings for a knot projection with a fixed
crossing, and give lower bounds for the warping degree.
