Four-genus and four-dimensional clasp number of a knot

For a knot $K$ in the $3$-sphere, by using the linking form on the first homology group of the double branched cover of the $3$-sphere, we investigate some numerical invariants, $4$-genus $g^*(K)$, nonorientable $4$-genus $\gamma^*(K)$ and $4$-dimensional clasp number $c^*(K)$, defined from the four-dimensional viewpoint. T. Shibuya gave an inequality $g^*(K)\leq c^*(K)$, and asked whether the equality holds or not. From our result in this paper, we find that the equality does not hold in general.