Quantum invariants, knot concordance and unknotting

我的研究方向是低维拓扑和纽结理论。这是纯数学的一个非常活跃的领域，在过去的几十年里有了非凡的发展，例如范畴化量子不变量的出现。这是一个多方面的领域，多种影响交织在一起。节点自然与三维拓扑中的概念相关，例如双曲几何或接触几何。人们可以从代数拓扑学的角度来看待纽结；这给了群论(通过基本群)和二次型理论(通过同调)的联系。纽结图--纽结在平面上的投影--本质上是装饰过的平面图，允许组合和图论技术在纽结理论中的应用。它们还导致了与量子代数的联系。除了它们与3-流形的关系外，纽结还与4-流形有着密切的联系，4-流形的光滑和拓扑范畴的二分性在纽结的水平上表现出来。我的主要目标是理解纽结理论不同部分之间的桥梁。这里有三个指导我研究的程序性问题：(1)塞弗特形式中包含什么拓扑信息？(2)量子不变量中包含什么几何信息？(3)纽结图揭示了3-空间中关于解结和曲面的什么？让我详细说明每一个目标。对于(1)，我使用了与二次型相关的三维和代数技术的混合，最终建立在弗里德曼的工作之上。我对相关问题做了一段时间的研究，得到了拓扑条亏格的一个有效上界。我的下一步将是完全代数刻画4-空间中补有无限循环基本群的拓扑曲面亏格。这样的拓扑量可以完全用代数的方式确定是相当罕见的。关于(2)，我将受益于多年的量子不变量，特别是Khovanov-Rozansky同调的经验。这是我工作的一个结果(将同调代数的方法与计算机计算相结合)，这些同调中的光滑协调信息比通常假设的要丰富得多，并且潜在地足够强，足以检测到协调群中的奇数扭转。分类的量子不变量形成了一个年轻而非常有活力的主题；我的工作针对该主题的核心问题之一，即不变量的几何解释。目标(3)触及了我刚刚开始研究的主题。然而，使用辫子处理和图论方法，我计划很快获得第一个结果：我将证明哪些规范曲面是拟正的，几乎正的结是强拟正的，以及正纤维结的解开数量与它们的亏格一致。这些陈述中的每一个都解决了一个公开的猜想。

My research is in low-dimensional topology and knot theory. This is a very active domain of pure mathematics, which has seen extraordinary development over the last decades, such as the advent of categorified quantum invariants. It is a multifaceted domain in which many influences mingle. Knots are naturally related to concepts from 3-dimensional topology, such as hyperbolic geometry or contact geometry. One can see knots from the perspective of algebraic topology; this gives a connection to group theory (via the fundamental group) and the theory of quadratic forms (via homology). Knot diagrams - projections of a knot to the plane - are essentially decorated plane graphs and allow for the application of combinatorial and graph theoretic techniques in knot theory. They also lead to a connection to quantum algebra. Aside from their relation to 3-manifolds, knots also enjoy close ties with 4-manifolds, and the dichotomy of the smooth and topological category of 4-manifolds manifests itself on the level of knots.My principal ambition is to understand the bridges between these different parts of knot theory. Here are three programmatic questions guiding my research:(1) What topological information is contained in the Seifert form?(2) What geometrical information is contained in quantum invariants?(3) What do knot diagrams reveal about unknotting and surfaces in 3-space?Let me elaborate on each of those goals. For (1), I am using a mixture of 3-dimensional and algebraic techniques relating to quadratic forms, ultimately building on Freedman’s work. I have worked on related questions for some time, and developed an effective upper bound for the topological slice genus. My next step will be to characterise completely algebraically the genus of topological surfaces in 4-space whose complement has infinite cyclic fundamental group. It is rather rare that such a topological quantity can completely be determined in an algebraic way.Regarding (2), I will profit from years of experience with quantum invariants, in particular Khovanov-Rozansky homologies. It is an upshot of my work (combining methods from homological algebra with computer calculation) that the smooth concordance information in those homologies is much richer than had been generally assumed, and potentially strong enough to detect odd torsion in the concordance group. Categorified quantum invariants form a young and very dynamic subject; my work is directed at one of the central questions of the subject, namely the invariants' geometrical interpretation.Goal (3) touches on a topic on which I have just begun to work. Nevertheless, using braid manipulation and graph theoretic methods I am planning to obtain first results soon: I am going to show which canonical surfaces are quasipositive, that almost positive knots are strongly quasipositive, and that the unknotting number of positive fibred knots agrees with their genus. Each of these statements resolves an open conjecture.