Critical point counts in knot cobordisms: abelian and metacyclic invariants

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.