Critical point counts in knot cobordisms: abelian and metacyclic invariants

For a pair of knots K 1 K_1 and K 0 K_0 , we consider the set of four-tuples of integers ( g , c 0 , c 1 , c 2 ) (g, c_0,c_1, c_2) for which there is a cobordism from K 1 K_1 to K 0 K_0 of genus g g having c i c_i critical points of each index i i . We describe basic properties that such sets must satisfy and then build homological obstructions to membership in the set. These obstructions are determined by knot invariants arising from cyclic and metacyclic covering spaces.