  In this paper, we study stable equivalence of exotically knotted surfaces in
4-manifolds, surfaces that are topologically isotopic but not smoothly
isotopic. We prove that any pair of embedded surfaces in the same homology
class become smoothly isotopic after stabilizing them by handle additions in
the ambient 4-manifold, which can moreover assumed to be attached in a standard
way (locally and unknottedly) in many favorable situations. In particular, any
exotically knotted pair of surfaces with cyclic fundamental group complements
become smoothly isotopic after a same number of standard stabilizations -
analogous to C.T.C. Wall's celebrated result on the stable equivalence of
simply-connected 4-manifolds. We moreover show that all constructions of exotic
knottings of surfaces we are aware of, which display a good variety of
techniques and ideas, produce surfaces that become smoothly isotopic after a
single stabilization.
