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

TL;DR

This paper demonstrates the existence of split, orientable 2-component surface-links in 4-sphere with non-isotopic splitting spheres, revealing a fundamental difference from classical 3-dimensional link theory.

Contribution

It constructs explicit examples of split surface-links in $S^4$ with non-isotopic splitting spheres, showing new phenomena in 4-dimensional topology.

Findings

Existence of non-isotopic splitting spheres in $S^4$ complements

Contrast with classical link theory in $S^3$

Explicit examples for $m,n ext{ with } m ext{,}n ext{ } extgreater= 4$

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 $S^3\setminus L$.