Studies in Knot Theory

PI: Joan S. birman我们建议研究与节和to3流形有关的几个问题：(1)我们希望利用标准接触结构在三维空间中找到新的Legendrian节和横向节的不变量。每个横结都可以表示为一个封闭的辫状结构，我们首先寻求对横结经典马尔可夫定理的适当修正。我们希望将Eliashberg和Fraser的工作推广到更广泛的节类，这表明横向节是由它们的结类型和Bennequin数决定的。在那之后，我们希望研究我们推测其定理不成立的结。我们有候选人。(2)我们提出研究解结识别算法问题的复杂性。我们推测多项式时间算法的存在性。(3)我们提出研究封闭的、可定向的3流形的Heegaard分裂的不变量，这些3流形是由曲面的映射类群表示到各种（最初是有限的）群上而产生的。一个这样的群集合是通过Wright的工作和Masbaum关于映射类群的Reshitikhan-Turaev表示的相关工作产生的。地球就是一个二维流形的例子。如果你站在地球上的任意两点上（对数学家来说，这两点被认为是没有地理特征的，而且是完美的圆形和光滑的），那么你的周围环境在这两点上看起来是一样的，并且没有给你任何关于全局拓扑结构的暗示。当人们了解到地球实际上是圆的，即它是一个二维流形，特别是它是一个二维球体时，人们对地球的看法发生了重大飞跃。但如果我们再增加一个维度，我们就处于哥伦布发现之前的人们的位置：一个点的邻域看起来和另一个点的邻域一样，我们对全球的图景一无所知。我们只知道世界是一个三维流形。它可能是一个三维球体，但也有许多其他的可能性，黑洞的存在表明了一个非常复杂的结构。这项工作是研究三维流形的最容易接近的例子，即当你从三维球体中移除一个打结的圆时留下的空间。结点是非常复杂的对象，对于3流形几何和拓扑来说，理解它们是非常基础的。通过寻找算法识别单个结点的方法，以及找到区分它们的部分方法，数学家们深入了解了拓扑结构的深层问题。我们还将研究其他三维流形，将它们切割成两个手柄体，它们沿着所谓的Heegaard表面粘贴在一起。

Proposal: DMS-9973232 PI: Joan S. BirmanWe propose to investigate several problems relating to knots and to3-manifolds: (1) We hope to find new invariants of Legendrian and transverse knots in 3-space, using the standard contact structure. Every transverse knot can be represented as a closed braid, and we begin by seeking an appropriate modification of the classical Markov theorem for transverse knots. We hope to extend the work of Eliashberg and Fraser, which shows that transverse knots are determined by their knot type and Bennequin number, to a wider class of knots. After that we hope to study knots for which we conjecture that their theorem fails. We have candidates. (2) We propose to study the complexity of the problem of recognizing the unknot algorithmically. We conjecture the existence of a polynomial-time algorithm. (3) We propose to study invariants of Heegaard splittings of closed, orientable 3-manifolds which arise through representations of the mapping class group of a surface onto various(initially finite) groups. One such collection of groups arises through the work of Wright and the related work of Masbaum on the Reshitikhan-Turaev representations of the mapping class group. The planet earth is an example of a 2-dimensional manifold. If you stand at any 2 points on the earth (which, to a mathematician, is to be thought of as having no geographical features and being perfectly round and smooth), then your surroundings look identical at the 2 points, and give you no hint of the global topology. A major leap in thinking about the earth occurred when it was learned that in fact the earth is round, i.e. it is a 2-dimensional manifold, in particular it is a 2-sphere. But if we add one more dimension, we are in the position of people before Columbus made his discovery: A `neighborhood of one point looks just like a neighborhood of another, and we have no idea of the global picture. All we know is that the world is a 3-dimensional manifold. It could be a3-dimensional sphere, but there are many other possibilities too, and the existence of black holes suggests a very complicated structure. The work is this proposal studies the most accessible examples of 3-dimensionalmanifolds, namely the space which remains when one removes a knotted circle from a 3-dimensional sphere. Knots are incredibly complicated objects, and understanding them is very basic to 3-manifold geometry and topology. By finding ways to recognize individual knots algorithmically, and by finding partial ways to tell them apart, mathematicians gain insight into deep questions about topology. We will also study other3-dimensional manifolds by cutting them apart into two handlebodies which are pasted together along a so-called Heegaard surface.