论文标题
图形配置的链接和3个manifolds的不变性
Invariants of links and 3-manifolds from graph configurations
论文作者
论文摘要
在这本独立的书中，遵循爱德华·维滕（Edward Witten），马克西姆·肯特维奇（Maxim Kontsevich），格雷格·库珀伯格（Greg Kuperberg）和迪伦·瑟斯顿（Dylan Thurston），我们定义了理性同源性3 spheres中框架链接的不变Z，我们研究了其属性。不变的Z通常称为Chern-Simons理论的扰动膨胀，在Jacobi图产生的分级空间中值。它计算了这种单位数图在环境歧管中的嵌入，从本书中解释的某种意义上，使用配置空间上的积分来解释，或者以双重方式将代数相交在同一配置空间中。当环境歧管是标准的3-Sphere时，不变的Z是由许多作者进行的通用Vassiliev链接不变性的，包括Guadagnini，Martellini和Mintchev，Bar-Natan，Bott and Bott and Taubes，Altschüler和Freidel，Thurston和Poirier ...这本书包含了这个更加灵活的Invariant的定义。我们将Z扩展到有理同源性圆柱体中的一类框架缠结类别的函子，并在包括某些电缆操作在内的各种操作下描述了该函子的行为。我们还计算了与有限类型不变的主要理论相关的离散衍生物的扩展不变式的迭代衍生物。与Massuyeau和Moussard的最新结果一起，我们的计算表明，Z限制为理性同源性3-Spheres（配备空链路）包含与这些流形的Le-Murakami-Ohtsuki LMO不变性相同的信息。他们还暗示了Z的一部分是卡森 - 骑行者不变的。
In this self-contained book, following Edward Witten, Maxim Kontsevich, Greg Kuperberg and Dylan Thurston, we define an invariant Z of framed links in rational homology 3-spheres, and we study its properties. The invariant Z, which is often called the perturbative expansion of the Chern-Simons theory, is valued in a graded space generated by Jacobi diagrams. It counts embeddings of this kind of unitrivalent graphs in the ambient manifold, in a sense that is explained in the book, using integrals over configuration spaces, or, in a dual way, algebraic intersections in the same configuration spaces. When the ambient manifold is the standard 3-sphere, the invariant Z is a universal Vassiliev link invariant studied by many authors including Guadagnini, Martellini and Mintchev, Bar-Natan, Bott and Taubes, Altschüler and Freidel, Thurston and Poirier... This book contains a more flexible definition of this invariant. We extend Z to a functor on a category of framed tangles in rational homology cylinders and we describe the behaviour of this functor under various operations including some cabling operations. We also compute iterated derivatives of our extended invariant with respect to the discrete derivatives associated to the main theories of finite type invariants. Together with recent results of Massuyeau and Moussard, our computations imply that the restriction of Z to rational homology 3-spheres (equipped with empty links) contains the same information as the Le-Murakami-Ohtsuki LMO invariant for these manifolds. They also imply that the degree one part of Z is the Casson-Walker invariant.
