  We study three knot invariants related to smoothly immersed disks in the
four-ball. These are the four-ball crossing number, which is the minimal number
of normal double points of such a disk bounded by a given knot; the slicing
number, which is the minimal number of crossing changes to a slice knot; and
the concordance unknotting number, which is the minimal unknotting number in a
smooth concordance class. Using Heegaard Floer homology we obtain bounds that
can be used to determine two of these invariants for all prime knots with
crossing number ten or less, and to determine the concordance unknotting number
for all but thirteen of these knots. As a further application we obtain some
new bounds on Gordian distance between torus knots. We also give a strengthened
version of Ozsvath and Szabo's obstruction to unknotting number one.
