Heegaard Floer homology of Matsumoto's manifolds

We consider a homology sphere Mn(K1,K2) presented by two knots K1, K2 with linking number 1 and framing (0,n). We call the manifold Matsumoto's manifold. We show that Mn(T2,3,K2) never bounds any contractible 4-manifold if n<2τ(K2) holds. We also give a formula of Ozsváth-Szabó's τ-invariant as the total sum of the Euler numbers of the reduced filtration. We compute the δ-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 new examples having a gap between what a knot is slice and what a double branched cover bounds a rational homology 4-ball.