Tunnel Number 1 Knots

工作的主要重点是在结理论领域，特别是隧道1号结的类别，或等效的结的外观承认属2 heegard分裂。这些包括许多常见类型的结，如2桥结、环面结和第11属的1桥结。已经在进行的工作给出了这类新的理论描述，通过将其与起源于群论的组合结构和手体映射类群的理论联系起来。这一描述产生了一个独特的程序来构建任何结隧道，甚至是所有隧道1号结的所有隧道的数值参数化。它为这一领域的研究提供了一个新的水平，许多方向正在进行的研究。与其他几位研究人员的额外工作将研究3流形的最小三角剖分问题，至少在初始阶段将开发软件来检查大量的例子。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，几十年来，三维拓扑学一直是一个蓬勃发展的研究领域。结的研究是其中心主题之一，并反映了这种丰富多样的观点。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，这些复合体是最近在低维拓扑和Teichmuller理论中非常感兴趣的对象。作为纯理论学科的基础研究，这项工作并不设想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。迄今为止的工作以及计划中的后续工作都需要PI的博士生大量参与。PI担任了7年他所在部门的研究生项目主任，特别强调招收女性和少数民族进入研究生水平的数学。PI还指导本科生的研究，并长期活跃于美国数学协会的区域活动，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.