  Given a connected cobordism between two knots in the 3-sphere, our main
result is an inequality involving torsion orders of the knot Floer homology of
the knots, and the number of local maxima and the genus of the cobordism. This
has several topological applications: The torsion order gives lower bounds on
the bridge index and the band-unlinking number of a knot, the fusion number of
a ribbon knot, and the number of minima appearing in a slice disk of a knot. It
also gives a lower bound on the number of bands appearing in a ribbon
concordance between two knots. Our bounds on the bridge index and fusion number
are sharp for $T_{p,q}$ and $T_{p,q}\# \overline{T}_{p,q}$, respectively. We
also show that the bridge index of $T_{p,q}$ is minimal within its concordance
class.
  The torsion order bounds a refinement of the cobordism distance on knots,
which is a metric. As a special case, we can bound the number of band moves
required to get from one knot to the other. We show knot Floer homology also
gives a lower bound on Sarkar's ribbon distance, and exhibit examples of ribbon
knots with arbitrarily large ribbon distance from the unknot.
