  We consider the set of connected surfaces in the 4-ball with boundary a fixed
knot in the 3-sphere. We define the stabilization distance between two surfaces
as the minimal $g$ such that we can get from one to the other using
stabilizations and destabilizations through surfaces of genus at most $g$.
Similarly, we consider a double point distance between two surfaces of the same
genus, which is the minimum over all regular homotopies connecting the two
surfaces of the maximal number of double points appearing in the homotopy. To
many of the concordance invariants defined using Heegaard Floer homology, we
construct an analogous invariant for a pair of surfaces. We show that these
give lower bounds on the stabilization distance and the double point distance.
We compute our invariants for some pairs of deform-spun slice disks by proving
a trace formula on the full infinity knot Floer complex, and by determining the
action on knot Floer homology of an automorphism of the connected sum of a knot
with itself that swaps the two summands. We use our invariants to find pairs of
slice disks with arbitrarily large distance with respect to many of the metrics
we consider in this paper. We also answer a slice disk analogue of Problem
1.105 (B) from Kirby's problem list by showing the existence of non-0-cobordant
slice disks.
