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

Abstract

For a knot K K in the 3 3 -sphere, by using the linking form on the first homology group of the double branched cover of the 3 3 -sphere, we investigate some numerical invariants, 4 4 -genus g ∗ ( K ) g^*(K) , nonorientable 4 4 -genus γ ∗ ( K ) \gamma ^*(K) and 4 4 -dimensional clasp number c ∗ ( K ) c^*(K) , defined from the four-dimensional viewpoint. T. Shibuya gave an inequality g ∗ ( K ) ≤ c ∗ ( K ) 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.