Critical point counts in knot cobordisms: abelian and metacyclic invariants

Abstract

For a pair of knots K 1 and K 0 , we consider the set of four-tuples of integers (g, c 0 , c 1 , c 2 ) for which there is a cobordism from K 1 to K 0 of genus g having c i critical points of each index 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.