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.
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