Tunnel Numbers, Heegaard Genus and Generalized Primality

隧道数、Heegaard 属和广义素性

• 批准号: 9803826
• 负责人: Jennifer Schultens
• 金额: $ 7.56万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-15 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803826&HistoricalAwards=false
• 关键词: Tunnel; Numbers; Heegaard; Genus; Generalized

9803826 Schultens This research project concerns the relation of Heegaard splittings to manifold decompositions, a relation that has been studied in a particular case under the guise of tunnel numbers of knots. The tunnel number of a knot has been shown to behave quite erratically under the operation of connected sum of knots. The project entails a more general investigation into the role of incompressible surfaces as they relate to Heegaard splittings. In particular, it should provide information on (1) lower bounds of the tunnel number of knots relative to their generalized prime decompositions; (2) bounds on the degeneration of tunnel number under connected sum of knots; (3) torus decompositions of manifolds; (4) the Heegaard genus of Haken manifolds; and (5) stabilization problems. The general philosophy of this project stems from a realization that 3-dimensional manifolds have been studied from several viewpoints. Two of these viewpoints involve cutting the 3-manifold along surfaces into basic building blocks. The idea of cutting along a surface to understand the 3-manifold in which the surface lies, is a natural extension of the idea of cutting along circles to understand the surface in which the circles lie. In the case of surfaces, the analysis of the circles contained in the surface suffices to characterize the surface. A complete classification has been obtained. 3-manifolds have proven to be far less tractable objects. In fact, extending the idea that has been so successful for the analysis of surfaces has engendered two very different viewpoints in the study of 3-manifolds. The investigator endeavors to reconcile these two viewpoints. ***

小行星9803826 这个研究项目涉及Heegaard分裂流形分解的关系，在一个特定的情况下，已经研究了隧道数的幌子下的结的关系。 纽结的隧道数在纽结的连通和运算下表现得相当不稳定。 该项目需要一个更一般的调查不可压缩表面的作用，因为它们涉及到Heegaard分裂。 特别是，它应该提供以下信息：（1）相对于广义素分解的纽结隧道数的下界;（2）纽结连通和下隧道数退化的界;（3）流形的环面分解;（4）哈肯流形的Heegaard亏格;（5）稳定化问题。 这个项目的一般理念源于对三维流形已经从几个角度进行了研究的认识。 这些观点中的两个涉及到将3-流形沿着表面切割成基本的构建块。 沿着沿着曲面切割以理解曲面所在的三维流形的思想，是沿着沿着圆切割以理解圆所在的曲面的思想的自然延伸。 在曲面的情况下，对曲面中包含的圆的分析足以表征曲面。 得到了一个完整的分类。 3-流形已经被证明是远不容易处理的对象。 事实上，扩展的想法，已经如此成功的分析曲面已经产生了两个非常不同的观点，在研究三维流形。 调查员努力调和这两种观点。 ***