  For pairs of knots K and J in the three-sphere, we consider the set of
four-tuples of integers (g,x,y,z) for which there is a cobordism from K to J of
genus g having x, y, and z, critical points of index 0, 1, and 2, respectively.
We describe basic properties that such sets must satisfy and then build
obstructions to membership in the set. These obstructions are based on
homological invariants arising from cyclic and metacyclic branched covering
spaces. A series of examples is presented. A concluding example demonstrates
that for each pair of integers g and n, there exists a ribbon knot K for which
any genus g cobordism from K to its reverse K^r must have at least n critical
points of each index.
