Development and categorification of a new link invariant

这个项目中的主要角色被称为链接。链接仅仅是3维空间中的结点的集合。这些结可以被认为是绑在非常细的绳子上的日常结。这项提议的广泛目标是研究一个新的链接不变量：斜率，这是我和两个合作者在2018年定义的。对链接的数学研究可以追溯到19世纪，卡尔·弗里德里希·高斯定义了链接积分，它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导致了彼得·格思里·泰特创造了第一个结表格。最终，纽结理论脱离了物理学，成为拓扑学这门新兴学科的一部分。如今，它又回来为其他科学提供信息。最近的发现为我们理解DNA和其他聚合物中打结现象的生物学相关性提供了线索。此外，通过拓扑量子计算的模型，纽结理论在量子计算机的构建中可能是至关重要的。具体地说，这个项目试图通过分离链路的一个组件来理解链路，让我们称之为K，并试图理解关于K如何与组成链路的其他节点相互作用的深层性质。为此，我们仔细观察K的一个小邻域，它在数学上是一个环面。这个圆环有两条重要的曲线，子午线和经度，我们的想法是理解这些曲线是如何位于空间中链接的补充物中的。现在，为了获得关于链接的更细微的信息，我们使用分支覆盖的数学构造，并在其中一个覆盖中查看纽结K和链接L的升降。斜率是一个与K有关的复数，我们在分支的封面上看到了它。这个数字有多种用途，允许人们写下关于其他经典链接不变量的强大公式，例如链接签名，正如我在下一段中解释的那样。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，它产生一个新的链接，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，如链接数或亚历山大多项式，现在已经完全理解了一段时间。然而，关于链接的签名在剪接下如何表现，文献中有一个令人惊讶的空白。正是在这种背景下，我和我的合作者介绍了链接的斜率。有了这个新的不变量，我们就能够写下人们长期寻找的两个环的拼接签名的公式。斜率的重要性远远超出了上面提到的签名公式。事实上，关于坡度仍有许多需要理解的地方，这将在低维拓扑学领域乃至更远的领域产生影响。在这个项目中，我们想要做的第一件事是获得一种快速计算该不变量的方法。然后，我们想把它与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一般化，我们将能够为更大类别的链接计算它们。最后，我们想对斜坡进行“分类”。这是一种数学结构，它采用了一个不变量，就像斜率一样，并发展了更丰富的理论，其中斜率只是一个很小的部分。坡度分类背后的想法是在拓扑量子场论的背景下进行的。一旦斜率被分类，我们将得到3-流形的不变量，而不是链环。这个不变量不仅对于研究三维流形，而且对于更好地理解拓扑量子场论都是非常重要的。

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.