  Let $LHT$ be a left handed trefoil knot and $K$ be any knot. We define
$M_n(K)$ to be the homology $3$-sphere which is represented by a simple link of
$LHT$ and $LHT \sharp K$ with framings $0$ and $n$ respectively. Starting with
this link, we construct homotopy $K3$ and spin rational homology $K3$ surfaces
containing $M_n(K)$. Then we apply the adjunction inequality to show that if
$n>2g^n_s(K)-2$, $M_n(K)$ does not bound any smooth spin rational $4$-ball, and
that under the same assumption the negative $n$-twisted Whitehead double of
$LHT \sharp K$ is not a slice knot, where $g^n_s(K)$ is the $n$-shake genus of
$K$.
