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.