Hyperbolic structures on knot complements
Hyperbolic structures on knot complements
复制标题
结补上的双曲结构
DOI：
10.1016/s0960-0779(97)00106-9
复制
发表时间：
1998
影响因子：
7.8
通讯作者：
A. Reid
中科院分区：
文献类型：
--
作者：
P. Callahan;A. Reid
在20世纪70年代中期3流形拓扑是革命性的思想瑟斯顿。主要推力瑟斯顿的想法是，几何结构存在于大多数3流形，这些几何可以用来研究拓扑结构的3流形。到目前为止，最复杂和最有趣的几何是双曲几何，双曲三维流形的研究已经成为三维流形拓扑学最近工作的焦点。本文的目的是调查的一些应用的存在性双曲结构的纽结补在S3。该调查是在大多数情况下自成一体，发展的援助下的例子，一些瑟斯顿的想法，并随后应用到纽结theory.The文件是组织如下;在第2节中，我们快速审查一些基本理论的结，他们的补充和基本群体。第2.3节包括一系列定义和一些术语，在阅读本文时，最好将本节作为参考。在第三节中我们回顾了一些双曲几何，在第四节中我们讨论了双曲三维空间中体积的计算。第5节讨论了纽结补集上的双曲结构，最后陈述了Thurston的显著定理（定理5.1）。剩下的部分，第6-10节，关注双曲结构存在性的应用。我们不要求对完整性，而是该文件只是打算调查一些许多应用程序的双曲几何3流形。
In the mid 1970s 3-manifold topology was revolutionized by the ideas of Thurston. The main thrust of Thurston’s ideas was that geometric structures existed on most 3-manifolds, and these geometries could be used to study the topology of 3-manifolds. By far the most complicated and interesting geometry is hyperbolic geometry, and the study of hyperbolic 3-manifolds has become a focal point for much recent work in 3-manifold topology. It is the intention of this paper to survey some of the applications of the existence of a hyperbolic structure on knot complements in S3. The survey is for the most part self-contained, developing with the aid of examples, some of Thurston’s ideas, and subsequent applications to knot theory.The paper is organized as follows; in Section 2 we quickly review some of the basic theory of knots, their complements and fundamental groups. Section 2.3 consists of a collection of definitions and some terminology and it may be best to treat this section as reference whilst reading the paper. In Section 3 we review some hyperbolic geometry, and in Section 4 we give some discussion on the computation of volume in hyperbolic 3-space. Section 5 discusses hyperbolic structures on knot complements, culminating in a statement of Thurston’s remarkable theorem (Theorem 5.1). The remaining sections, Sections 6-10, concern applications of the existence of a hyperbolic structure. We make no pretensions towards completeness, but rather the paper merely intends to survey some of the many applications of hyperbolic geometry in 3-manifolds.