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

本课题研究3-流形的量子不变量及其中的链接。一个主要的问题是，从拓扑和几何的角度来看，这些不变量的理解很差。这个项目的目标是在量子不变量和3流形的几何和拓扑的现有知识之间建立联系。主要的方法是研究闭3流形的量子不变量的大水平渐近展开。研究这些渐近的思想源于Witten对量子不变量的研究。从这个工作和数学家的后续工作中，出现了一些关于这些渐近的几何内容的严格的数学猜想。这些猜想表明量子不变量隐藏了深层拓扑信息。PI计划与他人合作，通过具体的案例研究来检验（一些）猜测。例：他将与J.E. Andersen一起继续他们的8字结手术项目。除了有限数量的3-流形外，所有这些3-流形都是双曲的，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3流形的体积。此外，PI将尝试从更一般的角度出发，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3-流形的不变量的渐近展开，吉田的构造是基于共形场论的阿贝尔化。在这方面，PI将与吉田共同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来获取低维物体（如结和三维空间）的形状（拓扑）和几何知识。我们应该研究低维空间及其几何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“结”结构，新的研究指出，人们可以应用结不变量来获得关于这种分子的信息。它们的某些特性取决于它们的形状，例如，它们如何“打结”。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？当我们在地球上行走时，我们应该想到这样一个事实，它看起来就像一个普通的平面，但事实上，地球的表面就像一个非常大的球的表面。这与我们的宇宙相似。局部一切看起来都是平的，也就是说，局部我们的宇宙看起来像一个标准的三维欧几里得空间（一个平面上的三维垂坠），但也许宇宙作为一个整体是完全不同的，就像一个弯曲的紧致空间。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，它们是否像球体一样弯曲或不像平面。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.琼斯在80年代发现的琼斯多项式。

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.