Critical point counts in knot cobordisms: abelian and metacyclic invariants

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