Searching for slice-ribbon counterexamples

寻找切片色带反例

• 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

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.

据说原始人相信地球是平的。这是基于当地数据的合理假设。低维拓扑学家研究3维和4维空间，称为流形，以了解我们居住的物理宇宙和时空的可能几何和拓扑。在这个项目中，结是三维空间中的一条光滑的闭合曲线--一个环。纽结是出现在许多科学领域的基本对象。特别地，它们是低维拓扑的构建块：每个3维或4维流形都可以用包含一个或多个结的柯比图来描述。理解四维空间中以给定纽结为边界的表面也是四维拓扑学的基础，并且与自然界中的线场（如磁力线和流体涡旋）的研究密切相关。这个雄心勃勃的项目将开发并在计算机程序中实现一种在四维空间中生成纽结二维球体的新方法，以及那些二维球的切片，它们是三维空间中的打结圆，它们在四维空间中绑定二维圆盘。福克斯提出的一个长期存在的猜想预测，这种片结实际上束缚了一种特殊的圆盘，称为带状圆盘，可以在三维空间中看到。我们的目标是解决这个猜想，找到一个切片结，不能绑定带状磁盘。