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1	Contributions to Quandle Theory: A Study of f-Quandles, Extensions, and Cohomology Churchill, Indu Rasika U. 19 May 2017 (has links) Quandles are distributive algebraic structures that were introduced by David Joyce [24] in his Ph.D. dissertation in 1979 and at the same time in separate work by Matveev [34]. Quandles can be used to construct invariants of the knots in the 3-dimensional space and knotted surfaces in 4-dimensional space. Quandles can also be studied on their own right as any non-associative algebraic structures. In this dissertation, we introduce f-quandles which are a generalization of usual quandles. In the first part of this dissertation, we present the definitions of f-quandles together with examples, and properties. Also, we provide a method of producing a new f-quandle from a given f-quandle together with a given homomorphism. Extensions of f-quandles with both dynamical and constant cocycles theory are discussed. In Chapter 4, we provide cohomology theory of f-quandles in Theorem 4.1.1 and briefly discuss the relationship between Knot Theory and f-quandles. In the second part of this dissertation, we provide generalized 2,3, and 4- cocycles for Alexander f-quandles with a few examples. Considering “Hom-algebraic Structures” as our nutrient enriched soil, we planted “quandle” seeds to get f-quandles. Over the last couple of years, this f- quandle plant grew into a tree. We believe this tree will continue to grow into a larger tree that will provide future fruit and contributions. Quandle f-Quandle Extension Cohomology Cocycle Mathematics
2	Generalizations of Quandles and their cohomologies Green, Matthew J. 05 July 2018 (has links) Quandles are distributive algebraic structures originally introduced independently by David Joyce and Sergei Matveev in 1979, motivated by the study of knots. In this dissertation, we discuss a number of generalizations of the notion of quandles. In the first part of this dissertation we discuss biquandles, in the context of augmented biquandles, a representation of biquandles in terms of actions of a set by an augmentation group. Using this representation we are able to develop a homology and cohomology theory for these structures. We then introduce an n-ary generalization of the notion of quandles. We discuss a number of properties of these structures and provide a number of examples. Also discussed are methods of obtaining n-ary quandles through iteration of binary quandles, and obtaining binary quandles from n-ary quandles, along with a classification of low order ternary quandles. We build upon this generalization, introducing n-ary f-quandles, and similarly discuss examples, properties, and relations between the n-ary structures and their binary counter parts, as well as low order classification of ternary f-quandles. Finally we present cohomology theory for general n-ary f-quandles. Cohomology Extension f-Quandle n-ary Quandle Operation Mathematics
3	Algebraické struktury pro barvení uzlů / Algebraic structures for knot coloring Vaváčková, Martina January 2018 (has links) Title: Algebraic Structures for Knot Coloring Author: Martina Vaváčková Department: Department of Algebra Supervisor: doc. RNDr. David Stanovský, Ph.D., Department of Algebra Abstract: This thesis is devoted to the study of the algebraic structures providing coloring invariants for knots and links. The main focus is on the relationship between these invariants. First of all, we characterize the binary algebras for arc and semiarc coloring. We give an example that the quandle coloring invariant is strictly stronger than the involutory quandle coloring invariant, and we show the connection between the two definitions of a biquandle, arising from different approaches to semiarc coloring. We use the relationship between links and braids to conclude that quandles and biquandles yield the same coloring invariants. Keywords: knot, coloring invariant, quandle, biquandle iii
4	On the Number of Colors in Quandle Knot Colorings Kerr, Jeremy William 22 March 2016 (has links) A major question in Knot Theory concerns the process of trying to determine when two knots are different. A knot invariant is a quantity (number, polynomial, group, etc.) that does not change by continuous deformation of the knot. One of the simplest invariant of knots is colorability. In this thesis, we study Fox colorings of knots and knots that are colored by linear Alexander quandles. In recent years, there has been an interest in reducing Fox colorings to a minimum number of colors. We prove that any Fox coloring of a 13-colorable knot has a diagram that uses exactly five colors. The ideas behind the reduction of colors in a Fox coloring is extended to knots colored by linear Alexander quandles. Thus, we prove that any knot colored by either the linear Alexander quandle Z5[t]/(t − 2) or Z5[t]/(t − 3) has a diagram using only four colors. Knot Theory Fox Coloring Linear Alexander Quandle Coloring Minimal Coloring Mathematics Physical Sciences and Mathematics
5	Quandle coloring conditions and zeros of the Alexander polynomials of Montesinos links / カンドル彩色条件とモンテシノス絡み目のアレキサンダー多項式の零点 Ishikawa, Katsumi 25 March 2019 (has links) 京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21536号 / 理博第4443号 / 新制\|\|理\|\|1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授大槻知忠, 教授向井茂, 教授小野薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM Alexander polynomial Montesinos link quandle coloring Hoste's conjecture alternating link 400
6	Objets tressés : une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles Lebed, Victoria 13 December 2012 (has links) (PDF) Dans cette thèse on développe une théorie générale des objets tressés et on l'applique à une étude de structures algébriques et topologiques. La partie I contient une théorie homologique des espaces vectoriels tressés et modules tressés, basée sur le coproduit de battage quantique. La construction d'un tressage structurel qui caractérise diverses structures - auto-distributives (AD), associatives, de Leibniz - permet de généraliser et unifier des homologies familières. Les hyper-bords de Loday, ainsi que certaines opérations homologiques, apparaissent naturellement dans cette interprétation. On présente ensuite des concepts de système tressé et module multi-tressé. Appliquée aux bigèbres, bimodules, produits croisés et (bi)modules de Hopf et de Yetter-Drinfel'd, cette théorie donne leurs interprétations tressées, homologies et actions adjointes. La no- tion de produits tensoriels multi-tressés d'algèbres donne un cadre unificateur pour les doubles de Heisenberg et Drinfel'd, ainsi que les algèbres X de Cibils-Rosso et Y et Z de Panaite. La partie III est orientée vers la topologie. On propose une catégorification des groupes de tresses virtuelles en termes d'objets tressés dans une catégorie symétrique (CS). Cette approche de double tressage donne une source de représentations de V Bn et un traitement catégorique des racks virtuels de Manturov et de la représentation de Burau tordue. On définit ensuite des structures AD dans une CS arbitraire et on les munit d'un tressage. Les techniques tressées de la partie I amènent alors à une théorie homologique des structures AD catégoriques. Les algèbres associatives, de Leibniz et de Hopf rentrent dans ce cadre catégorique. objet tressé homologie algébrique caractère module tressé algèbre de battage quantique complexe de Koszul homologie de quandle/rack homologie de Hochschild algèbre de Leibniz hyper-bord de Loday système tressé produit tensoriel multi-tressé produit croisé module de Yetter-Drinfel d (bi)module de Hopf double de Heisenberg double de Drinfel ' algèbre X R-matrice groupes de tresses virtuelles rack virtuel auto-distributivité catégorique représentation de Burau (tordue) structures auto-distributives libres

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