  We give a method for constructing many pairs of distinct knots $K_0$ and
$K_1$ such that the two 4-manifolds obtained by attaching a 2-handle to $B^4$
along $K_i$ with framing zero are diffeomorphic. We use the d-invariants of
Heegaard Floer homology to obstruct the smooth concordance of some of these
$K_0$ and $K_1$, thereby disproving a conjecture of Abe in [Abe16]. As a
consequence, we obtain a proof that there exist patterns $P$ in solid tori such
that $P(K)$ is not always concordant to $P(U) \# K$ and yet whose action on the
smooth concordance group is invertible.
