Studies in Knot Theory

提案：DMS-9973232 PI：Joan S. Birman我们打算研究与纽结和三维流形有关的几个问题：（1）我们希望利用标准切触结构找到三维空间中勒让德和横纽结的新的不变量。 每一个横结都可以表示为一个封闭的辫子，我们开始通过寻找一个适当的修正的经典马尔可夫定理的横结。 我们希望将Eliashberg和Fraser的工作扩展到更广泛的结类，他们的工作表明，横向结是由它们的结类型和Bennequin数决定的。之后，我们希望研究我们猜想他们的定理失败的结。 我们有候选人。 (2)我们建议研究的复杂性问题的识别unknot算法。我们猜想存在一个多项式时间算法。(3)我们建议研究不变量的Heegaard分裂的封闭，定向3-流形出现通过表示的映射类组的表面到各种（最初有限）的群体。 一个这样的群体的集合产生于赖特的工作和Masbaum对映射类群的Reshitikhan-Turaev表示的相关工作。 地球就是二维流形的一个例子。 如果你站在地球上的任何两个点上（对数学家来说，这两个点被认为是没有地理特征的，并且是完美的圆形和光滑的），那么你的周围环境在这两个点上看起来是相同的，并且没有给你任何全球拓扑结构的暗示。 一个重大的飞跃，在思考地球发生时，人们了解到，事实上地球是圆的，即它是一个二维流形，特别是它是一个2球。 但是，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就像另一个点的邻域，我们不知道全局图。 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。 这项工作是这项建议研究最容易的例子3-dimensionalmanifold，即空间仍然当一个删除一个打结的圆从一个3维领域。 纽结是非常复杂的对象，理解它们是三维几何和拓扑学的基础。 通过找到算法识别单个节点的方法，以及找到区分它们的部分方法，数学家们深入了解了拓扑学的深层问题。 我们还将研究其他三维流形，将它们切割成两个沿着沿着所谓的Heegaard曲面粘贴在一起的平行体。

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.