Three-Manifold Invariants: Towards the Property P Quest

DMS-0306774奥利弗·达斯巴赫能否利用纽结和同调三维球面的有限类型不变量来更深入地研究双曲体积或性质P的确定等经典问题，已成为低维拓扑学中的一个重要问题。本研究项目的目标之一是应用和推广三维流形不变量理论中使用的思想和方法，主要是纽结和同调球面的量子不变量，以取得性质P的进展。此外，这些方法与来自传奇理论和横向节点理论的工具相结合。其目的是得到结构上新的组合障碍，以确保纽结的性质P。反过来，从更经典的拓扑观点来看是自然的工具被应用于量子不变量的研究。认识到地球的形状是球体的形状，即球的表面，而不是圆盘，这是人类历史上的一大步。例如，另一种可能性是地球具有甜甜圈的表面形状。曲面是二维对象，称为二维流形。一个更高的维度，一个人必须处理类似的问题。对我们的三维宇宙可能是什么样子的理解要落后得多。三维拓扑学的目的是研究和分类整个宇宙可能被塑造的可能性。生成这些无限多个可能的3-流形的一种方法是通过打结的圆。人们通过在3-流形中取出一个小邻域的结点并以不同的方式插入它来得到一个新的3-流形。问题仍然存在，如何区分以这种方式出现的两种不同的3-流形。这项工作研究了结点的组合和几何性质对所得到的3-流形结构的影响。拓扑学已被证明是数学以外的领域应用的一种自然和非常富有成效的来源。例如，纽结理论在DNA函数的研究、密码学和量子计算模型的开发中都有应用。

DMS-0306774Oliver DasbachIt became an important question in low dimensional topology whether one can use finite type invariants of knots and homology 3-spheres to getmore insights into classical problems like the determination of the hyperbolic volume or the Property P quest. One of the goals of this research project is to apply and extend ideas and methods which are used in the theory of 3-manifold invariants, mainly quantum invariants for knots and homology spheres, towards progress in the Property P quest. Furthermore, these methods are combined with tools coming from the theory of Legendrian and Transversal knots. The aim is to get combinatorial obstructions, that are structurally new, ensuring Property P for a knot. In turn, tools that are natural from a more classical topological point of view are applied to the study of quantum invariants.It has been a huge step in the history of mankind to realize that theearth has the shape of a sphere, i.e. the surface of a ball, rather than it being a disk. Another possibility, for example, would have been that the earth has the shape of the surface of a doughnut. Surfaces are2-dimensional objects, called 2-manifolds. One dimension higher, one has to deal with similar problems. The understanding of how our 3-dimensional universe could look like is much less developed. The aim of three-dimensional topology is to study and classify the possibilities how the whole universe might be shaped. One way to generate these, infinitely many, possible 3-manifolds is through knotted circles. One gets a new 3-manifold by taking out a smallneighborhood of a knot in a 3-manifold and inserting it in adifferent way. The problem remains, how to distinguish two different 3-manifolds that arise that way. The work in this proposal studies the effect that combinatorial and geometric properties of the knot have on the structure of the resulting 3-manifold. Topology has been shown to be a natural and very fruitful source for applications in fields outside of mathematics. For example, knot theory has applications in the study of DNA functions, in cryptography and in the development of models for quantum computing.