The Geometry and Topology of Quantum Invariants of Knots and 3-Manifolds

结和3-流形的量子不变量的几何和拓扑

• 批准号: 0604994
• 负责人: David Auckly
• 金额: --
• 依托单位: Kansas State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604994&HistoricalAwards=false
• 关键词: Geometry; Topology; Quantum; Invariants; Knots

This project concerns the quantum invariants of 3-manifolds and links in them. A main problem is that these invariants are poorly understood from the point of view of topology and geometry. The goal of this project is to make connections between the quantum invariants and the existing knowledge of the geometry and topology of 3-manifolds. The main approach will be an investigation of the large level asymptotic expansion of the quantum invariants of closed 3-manifolds. The idea of studying these asymptotics stems from Witten's work on the quantum invariants. From that work and subsequent work of mathematicians there have emerged some rigorous mathematical conjectures about the geometric content of these asymptotics. These conjectures indicate that the quantum invariants hide deep topological information. The PI plans to check, in collaboration with others, (some of) the conjectures via concrete case studies. E.g., he will jointly with J.E. Andersen continue their project on the surgeries on the figure 8 knot. All but a finite number of these 3-manifolds are hyperbolic and it is the hope that this case study can shed some light on the connection between quantum invariants and hyperbolic geometry. A main question is whether the quantum invariants can detect the volume of hyperbolic 3-manifolds. In addition the PI will try to work from a more general point of view, namely he will apply T. Yoshida's new approach to the quantum invariants to try to calculate the asymptotic expansion of the invariants for all closed 3-manifolds, Yoshida's construction being based on an abelianization of conformal field theory. In that connection the PI will jointly with Yoshida work on proving that Yoshida's invariants coincide with the more well established Reshetikhin--Turaev invariants thereby making a connection to the other approaches to the quantum invariants.This project tries to develop new techniques to obtain knowledge about the shape (topology) and geometry of low-dimensional objects such as knots and 3-dimensional spaces. A main reason that we should study low-dimensional spaces and their geometry and shape is that nature contains such spaces in many contexts. Thus large molecules, e.g. DNA, have a "knotted" structure and newer research points in the direction that one can apply knot invariants to obtain information about such molecules. Certain of their properties depend on their shape, e.g., how they are "knotted". Another example is in cosmology where a central question is: What is the "shape" of our universe? Here one should think of the fact that when we walk around on the earth it just looks like an ordinary plane but in fact the surface of the earth is like the surface of a very large ball. It is similar with our universe. Locally everything looks flat, i.e., locally our universe looks like a standard 3-dimensional Euclidean space (a 3-dimensional pendant to a plane), but maybe the universe as a whole is something completely different, like a curved compact space. Mathematicians use so-called invariants to detect topological and geometric properties of spaces, e.g., if they are curved like a sphere or not like a plane. The quantum invariants, the theme for this project, is a relatively new family of invariants dating back to V. Jones' discovery of the Jones polynomial in the 80ties.

这个项目涉及三维流形的量子不变量和其中的链接。一个主要的问题是，这些不变量是从拓扑和几何的角度了解甚少。这个项目的目标是建立量子不变量和三维流形的几何和拓扑的现有知识之间的联系。主要的方法是研究封闭三维流形的量子不变量的大尺度渐近展开。研究这些渐近性的想法源于维滕对量子不变量的研究。从这项工作和随后的工作的数学家已经出现了一些严格的数学公式的几何内容，这些渐近。 这些结果表明量子不变量隐藏了深层的拓扑信息。PI计划与其他人合作，通过具体的案例研究来检查（一些）认证。例如，在一个示例中，他将与J. E.安德森继续他们的8字结手术项目。 除了有限数量的这些三维流形是双曲的，这是希望这个案例研究可以揭示量子不变量和双曲几何之间的联系。一个主要的问题是量子不变量是否可以检测双曲三维流形的体积。此外，PI将尝试从更一般的角度工作，即他将应用T。吉田的量子不变量的新方法，试图计算所有封闭的3-流形的不变量的渐近展开，吉田的建设是基于共形场论的阿贝尔化。在这方面，PI将与吉田共同努力，证明吉田的不变量与更完善的Reshetikhin-Turaev不变量相一致，从而与量子不变量的其他方法建立联系。该项目试图开发新技术，以获得有关低维物体（如结和三维空间）的形状（拓扑）和几何形状的知识。我们应该研究低维空间及其几何和形状的一个主要原因是，自然界在许多情况下都包含这样的空间。因此，大分子，例如DNA，具有“打结”结构，并且更新的研究指向可以应用结不变量来获得关于此类分子的信息的方向。它们的某些性质取决于它们的形状，例如，它们是如何“打结”的。另一个例子是在宇宙学中，一个中心问题是：我们宇宙的“形状”是什么？在这里，人们应该想到这样一个事实，当我们在地球上行走时，它看起来就像一个普通的平面，但实际上地球的表面就像一个非常大的球的表面。它与我们的宇宙相似。 局部看起来一切都是平的，即，我们的宇宙在局部上看起来像一个标准的三维欧几里得空间（一个平面的三维挂件），但也许宇宙作为一个整体是完全不同的东西，就像一个弯曲的紧凑空间。数学家使用所谓的不变量来检测空间的拓扑和几何性质，例如，如果它们像球体一样弯曲或者不像平面。量子不变量，这个项目的主题，是一个相对较新的不变量家族，可以追溯到V. Jones在80年代发现的琼斯多项式。