  The unknot U in S^4 has non-unique smooth spanning 3-balls up to isotopy
fixing U. Equivalently there are properly embedded non-separating 3-balls in
S^1xB^3 not properly isotopic to 1xB^3. More generally there exist
non-separating 3-spheres in S^1xS^3 not isotopic to 1xS^3 and non trivial
elements of \pi_0 Diff_0(S^1xS^3). Along the way we introduce barbell
diffeomorphisms, implantations and twistings to construct and modify
diffeomorphisms homotopic to the identity. We also introduce a 2-parameter
calculus of embeddings of the interval into 4-manifolds and introduce a framed
cobordism method as well as a direct method for showing that certain
2-parameter families are homotopically non trivial and diffeomorphisms are
isotopically nontrivial. Extensions to higher dimensional manifolds are
obtained.
