  An important difference between high dimensional smooth manifolds and smooth
4-manifolds that in a 4-manifold it is not always possible to represent every
middle dimensional homology class with a smoothly embedded sphere. This is true
even among the simplest 4-manifolds: $X_0(K)$ obtained by attaching an
$0$-framed 2-handle to the 4-ball along a knot $K$ in $S^3$. The $0$-shake
genus of $K$ records the minimal genus among all smooth embedded surfaces
representing a generator of the second homology of $X_0(K)$ and is clearly
bounded above by the slice genus of $K$. We prove that slice genus is not an
invariant of $X_0(K)$, and thereby provide infinitely many examples of knots
with $0$-shake genus strictly less than slice genus. This resolves Problem 1.41
of [Kir97]. As corollaries we show that Rasmussen's $s$ invariant is not a
$0$-trace invariant and we give examples, via the satellite operation, of
bijective maps on the smooth concordance group which fix the identity but do
not preserve slice genus. These corollaries resolve some questions from
[4MKC16].
