A Quantum Field Theory Approach to the Study of Low-dimensional Topology Invaraints and their Categorification

环的琼斯多项式不变量给出了两个谜题。第一个问题是在经典拓扑学的框架内对它的解释，以及它与连接补语的基本群的关系。这种关系在亚历山大多项式中广为人知，但它的琼斯类比却缺失了。我用量子场论的方法来寻找它。我将半经典极限中的有色Jones多项式分解为许多更简单的不变量之和，结果证明它们是链环的多元多项式不变量。这些不变量被自然地排列成一个“塔”，而亚历山大多项式就是它的基础。我的目标是研究其他“更高级别”多项式的性质，并建立它们与纽结补的拓扑的关系。琼斯多项式的第二个难题是它的多项式结构，这一点很难从其Chern-Simons-Witten路径积分表示中得出。关于琼斯多项式的多项式性质的一个令人惊讶的解释来自霍瓦诺夫的分类程序，该程序将琼斯多项式解释为分次模链复形的分次欧拉特征，该特征与一个环到同伦有关。我将研究霍万诺夫同调的可能的量子场论解释。也就是说，范畴复合体的同调应该是四维拓扑量子场论的态空间，这是必须构造的。这种构造有助于我们将Khovanov的结果推广到三维流形的量子多项式不变量，并解释它们与四维拓扑的关系。纽结的分类问题由来已久，尽管它的表述简单而透明，但尽管许多数学家共同努力，它仍然没有得到解决。纽结分类最成功的方法之一是构造纽结不变量，也就是数字，它可以通过查看纽结的图片很容易地计算出来，并且对于代表相同纽结的所有图片都是相同的。从琼斯多项式的发明开始，这一领域在过去的30年里经历了重大的发展。这个多项式和它后面的其他不变量与量子场论密切相关，可以用所谓的路径积分来表示。因此，纽结不变量理论中的许多新想法都受到量子物理方法和途径的启发。本研究的目的是利用量子场论的方法，在经典拓扑学的框架内解释新的“量子”拓扑不变量，并将它们应用于节点分类问题的求解。

The Jones polynomial invariant of links presents two puzzles. The first one is its interpretation within the framework of classical topology and its relation to the fundamental group of the link complement. This relation is well-known for the Alexander polynomial, but its Jones analog is missing. I search for it by using the quantum field theory approach. I decompose the colored Jones polynomial in the semi-classical limit into a sum of many simpler invariants, which turn out to be multi-variable polynomial invariants of links. These invariants are arranged naturally into a `tower' and the Alexander polynomial lies at its foundation. My goal is to study the properties of the other `higher level' polynomials and establish their relation to the topology of the knot complement. The second puzzle of the Jones polynomial is its polynomial structure, which does not follow easily from its Chern-Simons-Witten path integral presentation. An amazing explanation for the polynomial nature of the Jones polynomial comes from Khovanov's categorification program, which interprets the Jones polynomial as a graded Euler characteristic of a chain complex of graded modules, associated to a link up to a homotopy. I will study the possible quantum field theory interpretations of the Khovanov homology. Namely, the homology of the categorification complex should be a space of states for a 4-dimensional topological quantum field theory, which has to be constructed. This construction should help us to extend Khovanov's results to quantum polynomial invariants of 3-manifolds and to interpret their relation to 4-dimensional topology.The problem of knot classification is very old and although its formulation is simple and transparent, it has not been solved yet despite concerted efforts of many mathematicians. One of the most successful approaches to knot classification is the construction of knot invariants, that is, the numbers, which can be easily computed by looking at a picture of a knot and which would be the same for all pictures representing the same knot. This area has undergone significant developments in the last 30 years starting with the invention of the Jones polynomial. This polynomial and the other invariants which followed it, are intimately related to quantum field theory and can be expressed as so-called path integrals. Thus many of the new ideas in the theory of knot invariants are inspired by methods and approaches of quantum physics. The goal of my research is to use the methods of quantum field theory in order to interpret the new `quantum' topological invariants within the framework of classical topology and to apply them to the solution of the knot classification problem.