In a recent paper (Tatsuoka 2026), accepted by Inventiones Mathematicae, Tatsuoka proved that the unlink of two unknotted -spheres in admits infinitely many non-isotopic splitting -spheres.
We have discussed knots and links in several previous blog posts; see, for example, this post. The same ideas can be studied in higher dimensions by changing the dimension of the objects being knotted. For , the -sphere is where denotes the Euclidean norm. A -knot is a smoothly embedded copy of in . More generally, a -link is a finite disjoint union of smoothly embedded copies of in . Thus a two-component -link has the form where and are disjoint embedded -spheres in .
A -knot is called unknotted if it bounds a smoothly embedded -ball in . A two-component -link is called the unlink of two unknotted -spheres if and bound disjoint smoothly embedded -balls in .
For a -link , let denote a small open tubular neighbourhood of . For a single embedded -sphere, this is a small open normal thickening, homeomorphic to , where For a two-component link , the neighbourhood is the union of two such disjoint neighbourhoods. The complement is called the exterior of the link.
A splitting -sphere for a two-component -link is a smoothly embedded copy in the exterior such that and lie in different connected components of . Equivalently, every continuous path in from to must meet . Thus separates the two components of the link.
Two splitting -spheres and for the same link are called isotopic if they are isotopic through splitting -spheres in the link exterior. More explicitly, this means that there is a continuous family of smooth embeddings such that and each is a splitting -sphere for . A collection of splitting -spheres is called pairwise non-isotopic if no two distinct members of the collection are isotopic in this sense.
In classical knot theory, splitting spheres are rigid, in sense that for a two-component link in , any two splitting -spheres are isotopic in the link exterior. It is therefore natural to ask whether an analogous uniqueness statement holds one dimension higher. Dimension , however, often supports phenomena with no direct analogue in dimension . Hughes, Kim and Miller (Hughes et al. 2025) showed that splitting -spheres need not be unique for certain links of surfaces in , and they asked whether such non-uniqueness already appears for the simplest possible example: the unlink of two unknotted -spheres.
Tatsuoka answered this question affirmatively.
Theorem shows that even the most elementary two-component -link in has a surprisingly rich collection of separating -spheres in its exterior. The failure of uniqueness is not caused by knotting of the components: each component is an unknotted -sphere, and the two components are unlinked. Rather, the complexity comes from the many different ways in which -spheres can sit inside the complement while the link itself is kept fixed.