Tunnel Number 1 Knots

主要的工作集中在纽结理论领域，特别是隧道数为1的纽结，或者等价于其外部允许亏格为2的Heegaard分裂的纽结。这些结包括许多常见类型的结，如2桥结、环面结和1属1桥结。已经在进行的工作通过将它与起源于群论和手柄的映射类群的理论的组合结构联系起来，给出了这类的新的理论描述。该描述产生了构造任何结点隧道的独特过程，甚至产生了所有隧道编号1的所有隧道的所有隧道的数值参数。它为这一领域的新的调查水平提供了基础，目前正在进行的研究中正在寻求许多方向。与其他几名研究人员的额外工作将检查关于三维流形的最小三角剖分的问题，至少在其初始阶段将开发软件来检查大量实例。由于它与许多其他数学领域的联系，以及它与我们生活的三维空间的相关性，三维拓扑学几十年来一直是一个活跃的研究领域。对结的研究是它的中心主题之一，反映了这种丰富的观点多样性。这项正在进行的工作开发了隧道1号结与某些圆盘复合体和曲线复合体之间的新联系，这些复合体是低维拓扑学和泰希穆勒理论最近感兴趣的对象。作为纯理论学科的基础研究，这项工作不会立即应用于科学或技术。尽管如此，PI正在进行的研究项目在许多方面对教育和学生研究产生了更广泛的影响。到目前为止，这项工作及其计划的延续涉及到PI的博士生的大量参与。PI担任了他所在系研究生项目的七年主任，特别强调招募女性和少数族裔进入研究生水平的数学。PI还指导本科生研究学生，并长期活跃在美国数学协会的区域活动中，包括担任分会理事。

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.