Non-isotopic splitting spheres for a split link in $S^4$

Abstract

We show that there exist split, orientable, 2-component surface-links in $S^4$ with non-isotopic splitting spheres in their complements. In particular, for non-negative integers $m,n$ with $m\ge 4$, the unlink $L_{m,n}$ consisting of one component of genus $m$ and one component of genus $n$ contains in its complement two smooth splitting spheres that are not topologically isotopic in $S^4\setminus L_{m,n}$. This contrasts with link theory in the classical dimension, as any two splitting spheres in the complement of a 2-component split link $L\subset S^3$ are isotopic in $S^3\setminus L$.