Dehn Surgery on Links

小行星9803122 这个项目的目的是了解的拓扑结构的外部双曲结，承认一个非双曲德恩手术。 更具体地说，什么是双曲结的3-球，承认德恩手术是小塞弗特纤维空间或包含基本的2-球或2-环面？ 解决这一问题的主要方法是了解当一个小亏格的曲面，无论是本质曲面还是Heegaard曲面，可以出现在双曲纽结的Dehn手术中。 Dehn手术中表面的残留物被视为结外部的穿刺表面。 提出了通过观察这样一个穿刺表面与结内适当的平面的交点，可以获得结的良好图像。 这种方法在过去分析类似情况时是成功的。 三维空间的结构和结的分类是美丽而深刻的主题，但它们也是扩展我们在其他科学分支中的知识的重要主题。 例如，一些物理学家现在正试图将天文观测与三维空间的结构联系起来，以确定三维宇宙的形状。 微生物学家开始使用纽结理论的概念来描述和理解DNA分子的打结，化学家也在用同样的方法来理解其他分子的打结和连接。 Dehn手术构造是一种在三维空间上产生新的三维空间的操作。 这是通过改变周围的空间打结圈的空间。 这个项目研究了空间如何根据沿着变化的结的类型而变化。 结果空间与纽结之间的关系是复杂的，对它的理解是三维拓扑学和纽结理论的一个自然目标。 对这种关系的良好了解也是探索结和三维空间的有用工具。 自从在世纪之交引入以来，Dehn手术构造一直是在3维和4维空间中构造现象实例的标准方法。 它也证明了许多关于结的基本问题可以被公式化为关于Dehn手术构造的问题。***

9803122Luecke This project aims to understand the topology of the exterior of ahyperbolic knot that admits a non-hyperbolic Dehn surgery. Morespecifically, what are the hyperbolic knots in the 3-sphere that admitDehn surgeries that are small Seifert fibered spaces or contain essential2-spheres or 2-tori? The main approach to this problem is to understandwhen a surface of small genus, either an essential surface or a Heegaardsurface, can appear in a Dehn surgery on a hyperbolic knot. The residueof a surface in the Dehn surgery is seen as a punctured surface in theknot exterior. It is put forth that by looking at the intersections ofsuch a punctured surface with an appropriate planar surface in the knotexterior, one will obtain a good picture of the knot. Such an approachhas been successful in the past at analyzing similar situations. The structures of 3-dimensional spaces and the classification of knotsare beautiful and deep subjects, but they are also important subjects inextending our knowledge in other branches of the sciences. For example,some physicists now are trying to tie astronomical observations tostructures of 3-dimensional spaces in order to determine the shape of the3-dimensional universe. Microbiologists are beginning to use the conceptsof knot theory to describe and understand the knotting of DNA molecules.Chemists are doing the same to understand the knotting and linking of othermolecules. The Dehn surgery construction is an operation on a 3-dimensionalspace that produces a new 3-dimensional space. This is done by changingthe space around a knotted circle in the space. This project studies howthe space changes according to the type of knot along which the change wasmade. The relationship between the resulting space and the knot is acomplex one, and its understanding is a natural goal for 3-dimensionaltopology and knot theory. A good knowledge of this relationship is also auseful tool for exploring knots and 3-dimensional spaces. Since itsintroduction at the turn of the century, the Dehn surgery construction hasbeen a standard way of constructing examples of phenomena in 3- and4-dimensional spaces. It also turns out that many basic questions aboutknots can be formulated as questions about the Dehn surgery construction. ***