Knots, Heegaard Splittings and Width Complexes

这项拟议的研究源于更好地理解瘦位置的工作原理，无论是在纽结的背景下还是在三维流形的背景下。多年来，单纯复形、细胞复形和其他复形一直是几何群论中的重要特征。它们在三维流形研究中的作用是通过Masur和Minsky，Hempel等人的工作建立起来的。定义和研究“宽度复合体”将有助于简化目前对纽结理论和Heegaard分裂的研究。它还将重新阐述这些领域中的一些P.I.S宠物问题。其中一些问题与3-流形的亏格和阶在粘合下的行为有关。其他人则担心Heegaard分裂的稳定。还有一些人关注Heegaard分裂的不可缩性质。拟议的研究源于更好地理解纽结(即三维空间中的纽结圆)和三维流形(即曲面的三维推广)的愿望。这里的关键思想是使用某种类型的复合体，即一种奇特的记账设备，来编码关于结和3-流形的信息。有趣的是，这个复合体以新的视角重塑了研究领域的标准问题。该项目涉及面很广。其中一些部分对P.I.的研究生和前研究生开放。P.I.迄今已指导了三篇博士论文，并与西雅图大学的玛丽亚·罗宾逊(Maria Robinson)保持着特别密切的联系。私家侦探目前正在监督两名研究生：肖恩·拉尼尔和爱丽丝·史蒂文斯。私人情报局维持着几项国际合作。

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.