Abstract
In dimension 3, the following two theorems are well known: rst, any two Seifert surfaces bounded by the unknot are isotopic, and second, any two splitting spheres of the 2-component trivial link are isotopic. Recently, Budney-Gabai found that the generalization of the rst theorem to dimension 4 is false , i.e., there are in nitely many 3-balls bounded by the trivial 2-knot in the 4-sphere. In this talk, I will talk about the generalization of Budneyabai’s theorem to higher genus surface links. First, I will show that there exists a pair of handlebodies of genus greater than equal to two, bounded by the trivial surface knot in the 4-sphere, which are not isotopic rel. boundary. I will also talk about the exis- tence of 3-dimensional handlebody links in the 4-sphere that are not isotopic relative to the boundary by showing that there are non isotopic splitting spheres for trivial surface links. This talk is based on joint works with Mark Hughes and Maggie Miller.
Motivation
We take a look at Seifert-Fläche. How many seifert surfaces for a given knot inside the 3-sphere are there? For the unknot there is only one up to isotopy but for other knots there might be more. It turns out in 3D such knots do exist (Trotter, Alford, Lyon). In 4D Livingstone showed that the trivial link also only has one seifert surface. H.K.M.P. showed that there is a knot with two different seifert surfaces (whatever a seiert surface in 4D is)
More generally: Are there two handlebodies bounded by the same surface which are isotopically different. Hugh-K.-Miller showed that this is indeed the case. (Although you need to specify what it means to be bounded by the same surface first)
Questions
- What is a 4D-seifert surface?