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.
Update 22.09.2026. The paper (Tatsuoka 2026) has now been published. The reference is updated to the published version.