Searching for slice-ribbon counterexamples

寻找切片色带反例

基本信息

批准号：
EP/Y022939/1
负责人：
Brendan Edward Owens
金额：
$ 10.45万
依托单位：
University of Glasgow
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2023
资助国家：
英国
起止时间：
2023 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FY022939%2F1
关键词：
Searching slice ribbon counterexamples

项目摘要

Primitive people are said to have believed that the world was flat. This was a reasonable assumption based on local data. Low-dimensional topologists study 3- and 4-dimensional spaces known as manifolds in order to understand the possible geometry and topology of the physical universe and spacetime that we inhabit. For the purposes of this project, a knot is a smooth closed curve --- a loop --- in 3-dimensional space. Knots are fundamental objects which appear in many areas of science. In particular they are the building blocks for low-dimensional topology: every 3- or 4-dimensional manifold can be described using a Kirby diagram involving one or more knots. Understanding surfaces in 4-dimensional space whose boundary is a given knot is also fundamental for 4-dimensional topology, and is closely related to the study of line fields in nature such as magnetic field lines and fluid vortices.This ambitious project will develop, and implement in a computer program, a new approach to producing knotted 2-dimensional spheres in 4-dimensional space, and slices of those 2-spheres which are knotted circles in 3-dimensional space which bound 2-dimensional disks in 4-space. A longstanding conjecture due to Fox predicts that such slice knots in fact bound a special kind of disk called a ribbon disk which can be seen in 3-dimensional space. Our aim is to settle this conjecture by finding a slice knot which cannot bound a ribbon disk.

据说原始人认为世界是平坦的。这是基于本地数据的合理假设。低维拓扑师研究的3和4维空间被称为歧管，以了解我们所居住的物理宇宙和时空的可能几何形状和拓扑。出于该项目的目的，一个结是一个平滑的闭合曲线 - 在3维空间中的循环---。结是在许多科学领域出现的基本物体。特别是它们是低维拓扑的构件：可以使用涉及一个或多个结的柯比图来描述每3维或4维歧管。了解四维空间中的边界的表面，其边界也是4维拓扑的基础，与对自然界中的线场的研究密切相关，例如磁场线和流体涡流的研究。雄心勃勃的项目将在计算机上发展，并在计算机上进行实施，一种在计算机上生产的新方法，在这些方法中生产了开关的2个范围，该阶段在这些范围内，在这些范围内，在这些范围内，在这些范围内，在这些范围内，在这些范围内，在这些范围内，在这些范围内，在这些范围内，在这些范围内，在这些范围内，在这些范围内，在这些范围内融合了这些范围，并在这些范围内融合了这些旋转的范围，并在4二维领域中，并在2个范围内，以及2次旋转，并在2个范围内，以及2次旋转的方法3维空间以4空间结合2维磁盘。由于FOX而引起的长期猜想预测，这种切片的结实际上绑定了一种特殊的磁盘，称为带状盘，可以在三维空间中看到。我们的目的是通过找到无法结合丝带磁盘的切片结来解决这一猜想。