The intersection of three spheres in a sphere and a new application of the Sato-Levine invariant

Abstract

Take transverse immersions f from a disjoint unin of the three 4-spheres $S_1^4$, $S_2^4$, and $S_3^4$ into $S^6$ with the following properties: (1) The restriction of $f$ to $S_i^4$ is an embedding, (2) The intersection of $f(S_i^4)$ and $f(S_j^4)$ is not empty and connected, (3)The intersection among $f(S_1^4)$, $f(S_2^4)$, and $f(S_3^4)$ is not empty. Then we obtain three surface-links $L_i=(S_i^4\cap S_j^4,S_i^4\cap S_k^4)$ in $S_i^4$, where $(i,j,k)=(1,2,3),(2,3,1),(3,1,2).$ We prove that, we have the equality $\beta (L_1)+\beta (L_2)+\beta (L_3)=0$, where $\beta (L_i)$ is the Sato-Levine invariant of $L_i$, if all $L_i$ are semi-boundary links.

9pages, 1figure