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

Abstract:

Take transverse immersions $f:S^{4}_{1}\amalg$ $S^{4}_{2}\amalg$ $S^{4}_{3}\looparrowright S^{6}$ such that (1) $f\vert S^{4}_{i}$ is an embedding, (2) $f(S^{4}_{i})\cap f(S^{4}_{j})\neq \varnothing$ and $f(S^{4}_{i})\cap f(S^{4}_{j})$ is connected, and (3) $f(S^{4}_{1})\cap f(S^{4}_{2})\cap f(S^{4}_{3})$ $=\varnothing$. Then we obtain three surface-links $L_{i}$= ($f^{-1}(f(S^{4}_{i})\cap f(S^{4}_{j}))$, $f^{-1}(f(S^{4}_{i})\cap f(S^{4}_{k}))$ ) in $S^{4}_{i}$, 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.