Quantum invariants, knot concordance and unknotting

量子不变量、结一致性和解结

• 批准号: 412851057
• 负责人: Dr. Lukas Lewark
• 金额: --
• 依托单位: Department of Mathematics
• 依托单位国家: 德国
• 项目类别: Independent Junior Research Groups
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/412851057?language=en
• 关键词: Quantum; invariants; knot; concordance; unknotting

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.

我的研究方向是低维拓扑和纽结理论。这是纯数学中一个非常活跃的领域，在过去的几十年里取得了非凡的发展，例如分类量子不变量的出现。这是一个多方面的领域，许多影响交织在一起。纽结自然与三维拓扑学的概念有关，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;这给出了与群论（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-本质上是装饰平面图，并允许在纽结理论中应用组合和图论技术。它们还导致了与量子代数的联系。除了与三维流形的关系外，纽结也与四维流形有着密切的联系，四维流形的光滑和拓扑范畴的二分法在纽结的层次上表现出来。我的主要目标是理解纽结理论的这些不同部分之间的桥梁。这里有三个纲领性的问题指导我的研究：（1）什么拓扑信息包含在塞弗特形式？(2)量子不变量中包含什么几何信息？(3)纽结图揭示了三维空间中的解结和曲面的什么？请允许我详细阐述其中每一个目标。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。我研究相关问题已经有一段时间了，并为拓扑切片亏格建立了一个有效的上界。我的下一步将是完全代数地证明四维空间中补有无限循环基本群的拓扑曲面的亏格。关于（2），我将受益于多年的量子不变量的经验，特别是Khovanov-Rozansky同调。我的工作（将同调代数的方法与计算机计算相结合）的结果是，这些同调中的平滑一致性信息比通常假设的要丰富得多，并且可能足够强，可以检测一致性群中的奇扭转。分类的量子不变量是一门年轻而充满活力的学科;我的工作是针对这门学科的核心问题之一，即不变量的几何解释。目标（3）涉及我刚刚开始工作的一个主题。尽管如此，使用编织操纵和图论方法，我计划很快获得第一个结果：我将展示哪些典型的表面是准正的，几乎积极的结是强准正的，并且正的非结的数量与它们的属一致。这些陈述中的每一个都解决了一个公开的猜想。