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For pairs of knots K and J in the three-sphere, we consider the set offour-tuples of integers (g,x,y,z) for which there is a cobordism from K to J ofgenus 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 buildobstructions to membership in the set. These obstructions are based onhomological invariants arising from cyclic and metacyclic branched coveringspaces. A series of examples is presented. A concluding example demonstratesthat for each pair of integers g and n, there exists a ribbon knot K for whichany genus g cobordism from K to its reverse K^r must have at least n criticalpoints of each index.

Critical point counts in knot cobordisms: abelian and metacyclic invariants