  We consider a homology sphere $M_n(K_1,K_2)$ presented by two knots $K_1,K_2$
with linking number 1 and framing $(0,n)$. We call the manifold {\it
Matsumoto's manifold}. We show that there exists no contractible bound of
$M_n(T_{2,3},K_2)$ if $n<2\tau(K_2)$ holds. We also give a formula of
Ozsv\'ath-Szab\'o's $\tau$-invariant as the total sum of the Euler numbers of
the reduced filtration. We compute the $\delta$-invariants of the twisted
Whitehead doubles of torus knots and correction terms of the branched covers of
the Whitehead doubles. By using Owens and Strle's obstruction we show that the
$12$-twisted Whitehead double of the $(2,7)$-torus knot and the $20$-twisted
Whitehead double of the $(3,7)$-torus knot are not slice but the double
branched covers bound rational homology 4-balls. These are the first examples
having a gap between sliceness and rational 4-ball bound-ness of the double
branched cover.
