Tunnel Number 1 Knots

隧道 1 节

• 批准号: 0802424
• 负责人: Darryl McCullough
• 金额: $ 15.97万
• 依托单位: University of Oklahoma Norman Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0802424&HistoricalAwards=false
• 关键词: Tunnel; Number; Knots

The main focus of the work is in the area of knot theory, specifically the class of tunnel number 1 knots, or equivalently the knots whose exteriors admit genus-2 Heegaard splittings. These include many of the common types of knots, such as 2-bridge knots, torus knots, and genus-1 1-bridge knots. The work already underway gives a new theoretical description of this class, by relating it to combinatorial constructions originating in group theory and the theory of mapping class groups of handlebodies. This description yields a unique procedure to construct any knot tunnel, and even a numerical parameterization of all the tunnels of all tunnel number 1 knots. It provides the foundation for a new level of investigation in this area, with many directions being pursued in ongoing research. Additional work with several other investigators will examine questions about minimal triangulations of 3-manifolds, and at least in its initial stages will develop software to examine large collections of examples.Because of its connections with numerous other mathematical areas, and its relevance to the 3-dimensional space in which we live, 3-dimensional topology has been a vigorous area of research for many decades. The study of knots is one of its central themes, and reflects this rich diversity of viewpoints. The work underway develops new connections between tunnel number 1 knots and certain disk complexes and curve complexes, which are objects of much recent interest in low-dimensional topology and Teichmuller theory. As basic research in a pure theoretical discipline, the work does not envision immediate applications to science or technology. Nonetheless, there are numerous ways in which the PI's ongoing research program has a broader impact in education and student research. The work to date and its planned continuations involve heavy participation by the PI's doctoral students. The PI served for seven years as director of his department's graduate program, in particular stressing the recruitment of women and minorities into graduate-level mathematics. The PI also directs undergraduate research students, and has long been active in regional activities of the Mathematical Association of America, including service as Section Governor.

这项工作的主要重点是结理论领域，特别是隧道数字1节的类别，或者等效地将其外观允许的结属属于Heegaard分裂。其中包括许多常见的结类型，例如2桥结，圆环结和属1桥结。已经进行的工作通过将其与源自组理论的组合和映射手柄阶级的理论相结合的结构来提供了对该类别的新理论描述。此描述产生了一个独特的过程，可以构建任何结隧道，甚至是所有隧道数字1节的所有隧道的数值参数化。它为在这一领域进行新的调查水平奠定了基础，在正在进行的研究中有许多方向。与其他一些研究人员的其他工作将研究有关3个manifolds的最小三角剖分的问题，至少在其初始阶段，将开发软件来研究大量示例的集合。由于它与许多其他数学领域的联系，及其与3维空间的相关性，我们生活在三维拓扑的情况下，这是许多研究成年的剧烈研究领域。结的研究是其中心主题之一，反映了这一丰富的观点多样性。正在进行的工作开发了隧道数字1结与某些磁盘复合物和曲线复合物之间的新联系，这些磁盘复合体和曲线复合物是对低维拓扑和TeichMuller理论的最新兴趣的对象。作为纯粹的理论学科基础研究，该作品并未设想对科学或技术的立即应用。尽管如此，PI正在进行的研究计划对教育和学生研究产生了更大的影响。迄今为止的工作及其计划的延续涉及PI的博士学生的大量参与。 PI担任该部门研究生课程的董事七年，特别是强调将妇女和少数民族招募到研究生级数学中。 PI还指导本科生，并且长期以来一直积极参与美国数学协会的区域活动，包括服务部门的服务。