Framed links in 3-manifolds, its applications, and algebraic approaches to knot theory
Open AccessKnot theory is the study of embedded simple closed curves in 3-dimensional space. The field is substantial enough to study as a stand alone focus of research and encompasses various methods from other mathematical fields. Furthermore, there are deep connections between knot theory and numerous scientific fields such as: biology, chemistry, statistical mechanics, and physics. It is also a useful tool to study low-dimensional topology via Dehn surgery, Kirby Calculus, and invariants of 3-manifolds. This dissertation focuses on two approaches to knot theory, that is, geometric and algebraic. In the first two chapters we use framed links to study low-dimensional topology and the last two chapters are dedicated to studying knot theory in an algebraic setting. In particular, in the first chapter we discuss various important and classical topics in knot theory and end with an in-depth introduction to Dehn surgery, in the second chapter we study parallelizable manifolds and the change in framing of links via ambient isotopy, in the third chapter we focus on quandle homology, and in the fourth chapter we study Gram determinants in knot theory.The origin of the mathematical study of knots and links dates back to the 1800's and is largely due to Carl Friedrich Gauss. His collection of notebooks from the late 1700's to the mid 1800's contained various sketches of knots and links and methods of coding and distinguishing links. For example, in an entry of his notebook dated January 22, 1833 C. F. Gauss introduced the linking number of two knots as an integral. In 1910, Max Dehn created a technique for constructing ``Poincare spaces" which was later generalized to Dehn surgery in the work of R. H. Bing. In the 1960's, W. B. Raymond Lickorish and Andrew Hugh Wallace, separately and through different methods, proved that every closed orientable 3-manifold can be obtained from Dehn surgery along a link. This led to the study of low-dimensional topology through knot theory. In 1978, Robion Kirby described an equivalence relation on surgery descriptions generated by two moves called Kirby moves. The first chapter of my dissertation is made to be an introduction to knot theory, framed links, and introduces methods to studying low-dimensional topology through knot theory.The observation of the change in framing of framed links in 3-manifolds via ambient isotopy dates back to 1989 when Jozef Henryk Przytycki noticed that a change in framing via ambient isotopy was impossible for irreducible 3-manifolds. In 2003, Vladimir Chernov developed affine self-linking numbers by using Vassiliev-Goussarov invariants as a way to generalize the self-linking number. Through this generalization he was able to prove that it is impossible to change the framing of framed knots in a 3-manifold unless the 3-manifold contains a non-separating 2-sphere. In 2013, following Darryl McCullough's work on Dehn homeomorphism of 3-manifolds, Patricia Cahn, Vladimir Chernov, and Rustam Sadykov refined V. Chernov's theorem by proving that it is impossible to change the framing of framed knots in a 3-manifold unless the underlying unframed knot intersects a non-separating 2-sphere at exactly one point. In the second chapter of my dissertation I prove a generalization of their work (joint work with R. Bakshi, G. Montoya-Vega, J.H. Przytycki, and D. Weeks) to framed links; by showing that the only way of changing the framing of a framed link by ambient isotopy in an oriented 3-manifold is when the manifold admits a properly embedded non-separating 2-sphere and either the underlying unframed link intersects the non-separating 2-sphere at exactly one point or the underlying unframed link intersects the non-separating 2-sphere at exactly two points, each point belonging to a different component of the link. This change of framing is given by the Dirac trick, also known as the light bulb trick, which is illustrated in this chapter. The main tools used in the proof are based on D. McCullough's work on Dehn homeomorphisms and mapping class groups of 3-manifolds. In the same chapter I discuss parallelizable manifolds to complete a different proof, using even surgery, by A.T. Fomenko and S.V. Matveev, of E. Stiefel's theorem; 3-manifolds are parallelizable. Finally, to complete the proof of the main theorem of this chapter; we view spin structures as parallelizations of the tangent bundle of a 3-manifold to prove that the framing of a knot can only be changed via ambient isotopy by an even number of twists. The first homology theory related to a self-distributive structure was constructed in the early 1990s by Roger Fenn, Colin Rourke, and Brian Sanderson. They introduced the homology theory of racks motivated by higher dimensional knot theory. In 1998, J. Scott Carter, Seiichi Kamada, and Masahico Saito adapted the ideas to define the homology of quandles. One approach to studying quandle homology is to construct maps from the homology of quandles to that of groups. In 2010, Maciej Niebrzydowski and Jozef Henryk Przytycki constructed an isomorphism between Shur multipliers and the second quandle homology of a Takasaki quandle of an abelian group of odd order. In the third chapter of my dissertation I generalized M. Niebrzydowski and J.H. Przytycki's theorem (joint work with R. Bakshi, S. Mukherjee, T. Nosaka, and J.H. Przytycki) by expressing the second homology of an Alexander quasigroup quandle, in terms of exterior algebras. Additionally, this chapter presents a self-contained proof of its structure and provides a corollary relating the map used in the proof to quandle 2-cocycles. The main tools used in the proof are based on M. Niebrzydowski and J.H. Przytycki's work as well as algebraic methods.In the 1990's W.B. Raymond Lickorish introduced the Gram determinant of type A which is the determinant of a matrix given by a bilinear form on crossingless connections in the disk with 2n boundary points. This was motivated by his work on 3-manifold invariants to prove the existence and uniqueness of his construction of the Witten-Reshetikhin-Turaev invariants of 3-manifolds. Bruce Wallace Westbury independently constructed and proved a closed formula for the Gram determinant of type A while studying the representation theory of the Temperley-Lieb algebras related to statistical mechanics. Philippe Di Francesco also independently proved a closed formula while studying the meander problem related to combinatorics. Finally, Xuanting Cai gave a shorter proof of the closed formula than that of P. Di Francesco by utilizing the Jones-Wenzl idempotents to create a new basis for the Temperley-Lieb algebra. Also in the 1990's Paul Martin and Hubert Saleur formulated and proved a closed formula for the Gram determinant of type B while studying representation theory of an algebra, derived from the Temperley-Lieb algebra, that is associated to the transfer matrix formulation of statistical mechanics on arbitrary lattices. Qi Chen and J.H. Przytycki, motivated by the late Rodica Simion's work on chromatic joins, independently formulated and proved a closed formula to the Gram determinant of type B. The Gram determinant of type Mb was created in 2008 from the idea to work in the Mobius band, then in 2009 Qi Chen conjectured a general formula. In the fourth chapter of my dissertation I applied the theory of relative skein modules to generalize and refine the definition of Gram determinants in knot theory (joint work with R. Bakshi, S. Mukherjee, and J.H. Przytycki). Then I introduce the Gram determinant of generalized type A by using crossingless connections as the basis and a bilinear form similar to the bilinear form used in constructing type A; in particular the tangles are closed in the annulus instead of the 2-sphere. The main theorem in this chapter is a closed formula for the Gram determinant of generalized type A. The proof is an adaptation of X. Cai's proof of the Gram determinant of type A by changing the natural basis for the Temperley-Lieb algebra to a new basis consisting of tangles constructed with Jones-Wenzl idempotents and also generalizing many of X. Cai's lemmas to the annular case. The main tools used in the proof are Jones-Wenzl idempotents, theta nets, and mountain paths. We end the chapter by proving a few results that support Qi Chen's conjecture of a general closed formula for the Gram determinant of type Mb.
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