Titlebook: Lectures in Knot Theory; An Exploration of Co Józef H. Przytycki,Rhea Palak Bakshi,Deborah Weeks Textbook 2024 The Editor(s) (if applicable

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History of Knot Theory from Gauss to Jones,stance called ether. At this time, creating a table of the elements was of significant importance to the scientific community, and this theory encouraged Tait to work on the knot classification problem. In this lecture, the origins of knot theory are examined, taking as a starting point the developm

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From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology,ion into quandle invariants. Here we also describe the Wirtinger’s and Dehn’s presentations of the fundamental group of the link complement and their relation to Fox colorings. The second part of the lecture describes how the idea of Fox colorings can develop into the sophisticated notion of Yang-Ba

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Goeritz and Seifert Matrices,the checkerboard coloring of a link diagram and the second using an oriented surface bounded by a link. We discuss several link invariants coming from the matrix including the determinant, the signature, the Alexander-Conway polynomial, and the Tristram-Levine signature.

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The Jones Polynomial and Kauffman Bracket Polynomial,s lecture, we describe basic properties of these polynomials including mysterious relations with Fox 3??coloring. We also discuss Montesinos-Nakanishi 3??move conjecture and its solution using the Burnside group of link. We end by discussing the Nakanishi 4-move conjecture, from 1979.

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Variations on Catalan Connections and the Children Pairing Game,ith a simple interpretation of Catalan numbers with a topological flavor and presents a proof, motivated by the theory of skein modules, but which is very elementary and looks like a child’s game. We also discuss the lattice path and Dyck path interpretation of Catalan numbers (including Désiré Andr

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The Kauffman Bracket Skein Module and Algebra of Surface I-Bundles,ter varieties, cluster algebras, and quantum Teichmüller spaces. In this lecture we explore some of these connections and discuss the structure of the Kauffman bracket skein algebras of several thickened surfaces.

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