Three-Manifold Invariants: Towards the Property P Quest

在低维拓扑学中，是否可以利用纽结和同调3-球面的有限型不变量来深入研究经典问题，如双曲体积的确定或性质P的求取，已成为一个重要的问题.该研究项目的目标之一是应用和扩展在3-流形不变量理论中使用的思想和方法，主要是结和同调球的量子不变量，以实现性质P的探索。此外，这些方法结合了来自勒让德和transmandrian结的理论工具。目的是得到结构上新的组合障碍物，确保结的属性P。反过来，从更经典的拓扑学观点来看是自然的工具被应用于量子不变量的研究。认识到地球具有球形，即球的表面，而不是圆盘，这是人类历史上的一个巨大进步。例如，另一种可能性是地球的表面形状像甜甜圈。曲面是二维物体，称为二维流形。高一个维度，人们必须处理类似的问题。我们对三维宇宙的理解还远远不够。三维拓扑学的目的是研究和分类整个宇宙可能如何形成的可能性。生成这些无穷多个可能的三维流形的一种方法是通过打结的圆。通过在一个3-流形中取出一个纽结的一个小邻域，并以不同的方式插入它，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个不同的三维流形。在这个建议中的工作研究的效果，组合和几何性质的结的结构上产生的3-流形。拓扑已被证明是一个自然的和非常富有成效的来源应用领域以外的数学。例如，纽结理论在DNA功能的研究、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.