Knots, Heegaard Splittings and Width Complexes

结、Heegaard 裂口和宽度复合体

• 批准号: 0603736
• 负责人: Jennifer Schultens
• 金额: --
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603736&HistoricalAwards=false
• 关键词: Knots; Heegaard; Splittings; Width; Complexes

The proposed research grows out of a desire to better understand the workings of thin position, both in the context of knots and in the context of 3-manifolds. Simplicial complexes, cell complexes and other complexes have featured prominently in geometric group theory for many years. Their utility in the study of 3-manifolds has been established through the work of Masur and Minsky, Hempel and others. Defining and studying the ``width complex'' will help streamline current research on knot theory and Heegaard splittings. It will also reformulate some of the P.I.'s pet problems in these areas. Some of these problems pertain to the behavior of genus and rank under gluing of 3-manifolds. Others concern stabilizing of Heegaard splittings. Yet others concern the nature of untelescopings of Heegaard splittings.The proposed research grows out of a desire to better understand knots, that is, knotted circles in 3-dimensional space, and 3-manifolds, that is, 3-dimensional generalizations of surfaces. The key idea here is to use a certain type of complex, that is, a fancy bookkeeping device, to encode information about knots and 3-manifolds. Interestingly, this complex recasts standard problems of the research area in a new light. The project is extensive. Some parts are accessible to graduate students and former graduate students of the P.I. The P.I. has directed three Ph.D. dissertations to date and maintains especially close contact with Maria Robinson (Seattle University). The P.I. is currently supervising two graduate students: Shawn Lanier and Alice Stevens. The P.I. maintains several international collaborations.

提出的研究增长的愿望，以更好地了解薄的位置，无论是在结的上下文中，并在3-流形的上下文中的工作。 单纯复形、胞腔复形和其他复形在几何群论中有着突出的地位。 他们的效用在研究3-流形已建立通过工作的Masur和明斯基，亨佩尔和其他人。定义和研究“宽度复合体”将有助于简化当前对纽结理论和Heegaard分裂的研究。 它还将重新制定一些PI。在这些领域的宠物问题。 这些问题中的一些涉及的行为的亏格和秩下胶合的3-流形。 另一些则是关于稳定Heegaard分裂。还有一些人关注Heegaard分裂的非伸缩性的性质。拟议中的研究是出于更好地理解结（knots）和3-流形（3-manifold）的愿望，结是3维空间中的打结圆，3-流形是曲面的3维推广。 这里的关键思想是使用某种类型的复形，也就是一种奇特的簿记设备，来编码关于纽结和三维流形的信息。有趣的是，这个复杂的重铸标准问题的研究领域在一个新的光。 该项目是广泛的。 有些部分是可访问的研究生和前研究生的P.I. 私家侦探曾指导过三个博士学位他的论文至今，并保持特别密切的联系与玛丽亚罗宾逊（西雅图大学）。 私家侦探目前正在指导两名研究生：肖恩拉尼尔和爱丽丝史蒂文斯。 私家侦探保持了多项国际合作。