Knot and 3-manifold invariants and Dehn surgery

PI建议探索经典的三维拓扑技术在研究结和3-流形的“有限型”不变量中的适用性。她想研究不能用有限型不变量区分的纽结的内在不变量（例如纽结属、卫星结构）之间的关系。为此，她建议使用多年来发展起来的一般机制来理解手术下3-流形的拓扑如何变化。由此产生的工作将揭示如何有限型纽结不变量与纽结补的拓扑结构，并可能导致进展的开放问题，这些不变量是否区分所有的纽结。她还想寻找纽结的琼斯多项式与Dehn手术获得的3-流形的基本群之间的关系。该项目的研究领域在于三维拓扑学，其中心研究对象是称为3-流形的空间。三维流形是一种局部看起来像普通三维空间但其整体结构可能很复杂的对象。三维拓扑学的一个主要目标是理解这些结构并实现三维流形的分类。三维拓扑学的一个重要部分也是研究纽结（以某种纠缠的方式嵌入三维流形中的环）及其分类。拓扑学家处理这些问题的方法之一是使用“不变量”。近年来，源于物理学的思想引导数学家们发现了纽结和三维流形的各种不变量。PI项目的中心主题是利用传统三维拓扑学和物理学的思想来理解这些不变量的性质，并研究它们在多大程度上与纽结和三维流形相交。

The PI proposes to explore the applicability of classical 3-dimensional topology techniques in the study of the ``finite type" invariants of knots and 3-manifolds. She would like to investigate the relationbetween intrinsic invariants (e.g. knot genus, satellite structures) of knots that cannot be distinguished by their finite type invariants.For this she proposes to use the general machinery that has been developedover the years to understand how the topology of 3-manifolds changes under surgery. The resulting work will shed some light on how the finite type knot invariants relate to the topology of the knot complement and couldlead to progress on the open question of whether these invariantsdistinguish all knots. She would also like to search for relations between the Jones polynomial of knots and the fundamental group of the 3-manifoldsobtained by Dehn surgery on knots.The research of the project lies in the area of 3-dimensional topology thecentral objects of study of which are spaces called 3-manifolds. A3-manifold is an object that locally looks like the ordinary 3-dimensionalspace but whose global structure can be complicated. A main goal of3-dimensional topology is to understand these structures and achieve aclassification of 3-manifolds. An important part of 3-dimensional topologyis also the study of knots (loops embedded in some tangled way in3-manifolds) and their classification. One of the ways that topologists havebeen approaching these problems is through the use of ``invariants". In therecent years, ideas originated in physics, lead mathematicians to thediscovery of a variety of invariants of knots and 3-manifolds. The centraltheme of the PI's project is to understand the properties of theseinvariants, using ideas from traditional 3-dimensional topology and fromphysics, and investigate the extent to which they distinguishknots and 3-manifolds.