Development and categorification of a new link invariant

这个项目的主角被称为链接。一个连杆就是三维空间中结的集合。这些结可以被认为是日常结系在非常细的绳子上。该提案的总体目标是研究一个新的链接不变量：斜率，这是我和两位合作者在2018年定义的。链接的数学研究可以追溯到19世纪，卡尔·弗里德里希·高斯（Carl Friedrich Gauss）定义了链接积分，这是最早的链接不变量之一。19世纪60年代，开尔文勋爵关于原子是以太中的结的理论促使彼得·格斯里·泰特（Peter Guthrie Tait）创建了第一个结表。最终，结理论脱离了物理学，成为拓扑学新兴学科的一部分。如今，它又回来为其他科学提供信息。最近的发现使我们对DNA和其他聚合物中打结现象的生物学相关性有了更深入的了解。此外，通过拓扑量子计算模型，结理论可能对量子计算机的构建至关重要。具体来说，这个项目试图通过分离其中一个组件来理解一个链接，让我们称之为K，并试图理解K如何与构成该链接的其他节点相互作用的深层特性。为了达到这个目的，我们仔细观察K的一个小邻域，它在数学上是一个环面。这个环面有两条重要的曲线，子午线和经线，我们的想法是理解这些曲线是如何在空间的连接中互补的。现在，为了得到关于连杆的更细微的信息，我们使用分支连杆的数学结构，并观察其中一个连杆K和连杆L的提升。斜率是一个与K相关的复数我们在分支封面中读到过。这个数字有多种应用，允许人们写出关于其他经典链接不变量的强大公式，例如链接签名，我将在下一段中解释。给定两个链接，L1和L2，有一个重要的操作，称为L1和L2的拼接，产生一个新的链接，l。数学家试图理解L1和L2的不同性质在拼接操作下的表现。许多不变量的行为，如连接数或亚历山大多项式，已经被完全理解了一段时间。然而，有一个令人惊讶的差距，在文献中关于一个链接的签名如何表现在拼接。正是在这种背景下，我和我的合作者引入了链接的斜率。有了这个新的不变量，我们就能写出寻找已久的两个连杆拼接的签名公式。斜率的重要性远远超出了上面提到的签名公式。事实上，关于斜率还有很多需要理解的地方，这将对低维拓扑和其他领域产生影响。在这个项目中，我们要做的第一件事就是找到一种快速计算这个不变量的方法。然后，我们想把它和其他已知的不变量联系起来，比如科克伦导数和米尔诺数。这些联系很重要，因为斜率将给出这些经典不变量的泛化，我们将能够在更大的连杆类中计算它们。最后，我们要对斜率进行“分类”。这是一种数学结构，它采用了一个不变量，比如斜率，并发展了一个更丰富的理论，其中斜率只是很小的一部分。斜率的分类背后的思想是在拓扑量子场论的背景下。一旦斜率被分类，我们将得到3流形的不变量，而不是连杆。这个不变量不仅对研究3流形很重要，而且对更好地理解拓扑量子场论也很重要。

The main characters in this project are called links. A link is simply a collection of knots in the 3-dimensional space. These knots can be thought of as everyday knots tied on very thin strings. The broad goal of this proposal is to investigate a new link invariant: the slope, which I, together with two collaborators, defined in 2018. The mathematical study of links goes back to the 19th century with Carl Friedrich Gauss who defined the linking integral, one of the first link invariants. In the 1860s, Lord Kelvin's theory that atoms were knots in the aether led to Peter Guthrie Tait's creation of the first knot tables. Eventually knot theory moved away from physics and became part of the emerging subject of topology. Nowadays it has come back to inform other sciences. Recent discoveries have shed light into our understanding of the biological relevance of knotting phenomena in DNA and other polymers. Moreover, knot theory may be crucial in the construction of quantum computers, through the model of topological quantum computation.Concretely, this project tries to understand a link by isolating one of its components, let us call it K, and trying to understand deep properties about how K interacts with the other knots that make up the link. To this end we look closely at a little neighborhood of K, which is mathematically a torus. This torus has two important curves, the meridian and the longitude, and the idea is to understand how these curves sit in the complement of the link in the space. Now, to get more subtle information on the link, we use the mathematical construction of branched covers, and look at the lifts of the knot K and of the link L in one of these covers. The slope is a complex number associated to K which we read in the branched cover. This number has multiple applications, allowing one to write down powerful formulas about other classic link invariants, like for example the link signature, as I explain in the next paragraph.Given two links, L1 and L2, there is an important operation, called the splice of L1 and L2, which produces a new link, L. Mathematicians have tried to understand how different properties of L1 and L2 behave under the splicing operation. The behaviour of many invariants, such as linking numbers or the Alexander polynomial, has been completely understood for some time now. However, there was a surprising gap in the literature regarding how does the signature of a link behaves under splicing. It is in this context that my collaborators and myself introduced the slope of a link. With this new invariant in hand, we were able to write down the long-sought formula of the signature of the splice of two links. The importance of the slope goes far beyond the above mentioned formula for the signature. Indeed, there is still much to be understood about the slope, which will have consequences in the field of low dimensional topology and beyond. One of the first things we want to do in this project is to get a fast way to compute this invariant. Then, we want to relate it to other known invariants, like Cochran's derivatives and Milnor numbers. These connections are important since the slope will give a generalization of these classic invariants and we will be able to compute them for a much larger class of links. Finally, we want to 'categorify' the slope. This is a mathematical construction which takes an invariant, like the slope, and develops a much richer theory in which the slope is just a tiny part. The idea behind the categorification of the slope sits in the context of Topological Quantum Field Theories. Once the slope is categorified we will obtain an invariant of 3-manifolds, as opposed to links. This invariant will be important, not only to study 3-manifolds, but also to better understand the Topological Quantum Field Theory.