A Historical Exploration of Knot Theory, Khovanov Homology, and Framing Changes of Links and Skein Modules
Open AccessThe fascinating mathematical theory of knots studies the embeddings of simple closed curves, up to natural deformations, in the three dimensional space $\mathbb{R}^3$ (or in $\mathbb{S}^3=\mathbb{R}^3 \cup \infty$). Historically originated as a sub-field of topology, the theory underwent significant scientific advancements so that knot theory is currently a research field of its own. Nonetheless, the theory enjoys broad connections with other fields such as chemistry, biology, physics, and statistical mechanics.\\ One of the fundamental problems in the theory is the classification. This classification is done up to the natural movement in space which is called an ambient isotopy. In 1927, the German mathematician Kurt Reidemeister showed that two link diagrams, possibly oriented, are isotopic if and only if they are connected by a finite sequence of moves, called Reidemeister moves, and planar isotopy. Following this idea, to distinguish knots and links one has to look for invariants of links, that is, properties of links that remain unchanged under ambient isotopy. In other words, to show that a property $P(D)$ of a diagram $D$ is an invariant, one has to check that it is preserved under Reidemeister moves.\\ This dissertation explores knot theory beginning with a description of its early origins and developments up to the first displays of formalization in Chapter 1. Khovanov homology, one of the most powerful link invariants, is introduced and a result about the torsion in Khovanov homology of torus links is proved in Chapter 2. Furthermore, skein modules are defined and connected to an important result about framing changes of links in $3$-manifolds in Chapter 3.\\ Chapter 1 of the dissertation aims to provide a through history of the early origins of knot theory. One motivation is the fact that humanity's fascination with knots can be traced back to prehistoric times. The first part of this chapter explores the early appearances of knots and presents a chronological overview of the developments that contributed to the first building-blocks towards the formalization of the theory. As the reader may surmise, Ancient Greek mathematics played an important role in the initial stages. Some well-known historical names such as Leonardo da Vinci, Pacioli, D\"urer, Leibniz, Euler, and Gauss, also take part in the narrative. The second part of this chapter examines the developments that occurred in XIX century Scotland associated to the work of Maxwell, Tait, and Kelvin. Special attention is given to the now called Tait conjectures and the evolution of the classification of knots. Moreover, we stress how progress in other areas such as algebraic topology and graph theory, positively influenced the growth of knot theory. \\ Chapter 2 provides a construction of Khovanov homology. A powerful link invariant, this homology theory has been extensively studied since its introduction in the late 1990s by Mikhail Khovanov. The author presents a short historical account on how Khovanov realized the core of his homology theory idea, curiously at the JFK airport. Khovanov homology offers a nontrivial generalization of the Jones polynomial of links in $\mathbb{R}^{3}$. In this chapter, Khovanov homology is introduced as understood by Oleg Viro. Referred to as the unoriented framed version of Khovanov homology, this approach initially gives a categorification of the Kauffman bracket polynomial. Then, this categorification is connected to Khovanov's original construction. With the goal of illustrating such an elegant process, an example of Khovanov homology computation is presented. One of the most important aspects in studying Khovanov homology is understanding the torsion information in terms of topological properties of the link. The last part of the chapter shows a construction of the long exact sequence of Khovanov homology which is a categorification of the Kauffman skein relation. This long exact sequence is used to explicitly compute the Khovanov homology of torus links of the type $T(2,n)$. \\ The notion of a skein module was introduced in 1987 by Józef H. Przytycki as a generalization of polynomial link invariants in $S^3$ to arbitrary $3$-manifolds. Chapter 3 gives a historical introduction of skein modules, formally defines this structure, and provides some specific examples of computed skein modules. Subsequently, skein modules are connected with framing changes of links in $3$-manifolds. More precisely, special attention is given to the result that affirms that the only way of changing the framing of a knot or a link, by ambient isotopy in an oriented 3-manifold, is when the manifold has a properly embedded non-separating sphere. Moreover, this change of framing is given by the Dirac trick which is also known as the light bulb trick. In particular, we focus on the framing skein module, the $q$-homology skein module, and the Kauffman bracket skein module.
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