Dissertations / Theses on the topic 'Quandles'

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.

Quandles were introduced to Knot Theory in the 1980s as an almost complete algebraic invariant for knots and links. Like their more basic siblings, groups, they are difficult to distinguish so a major challenge is to devise means for determining when two quandles having different presentations are really different. This thesis addresses this point by studying algebraic aspects of quandles. Following what is mainly a recapitulation of existing work on quandles, we firstly investigate how a link quandle is related to the quandles of the individual components of the link. Next we investigate coset quandles. These are motivated by the transitive action of the operator, associated and automorphism group actions on a given quandle, allowing techniques of permutation group theory to be used. We will show that the class of all coset quandles includes the class of all Alexander quandles; indeed all group quandles. Coset quandles are used in two ways: to give representations of connected quandles, which include knot quandles; and to provide target quandles for homomorphism invariants which may be useful in enabling one to distinguish quandles by counting homomorphisms onto target quandles. Following an investigation of the information loss in going from the fundamental quandle of a link to the fundamental group, we apply our techniques to calculations for the figure eight knot and braid index two knots and involving lower triangular matrix groups. The thesis is rounded out by two appendices, one giving a short table of knot quandles for knots up to six crossings and the other a computer program for computing the homomorphism invariants.

This thesis arose from a desire to better understand the structures of automorphism groups and inner automorphism groups of quandles. We compute and give the structure of the automorphism groups of all dihedral quandles. In their paper Matrices and Finite Quandles, Ho and Nelson found all quandles (up to isomorphism) of orders 3, 4, and 5 and determined their automorphism groups. Here we find the automorphism groups of all quandles of orders 6 and 7. There are, up to isomoprhism, 73 quandles of order 6 and 289 quandles of order 7.

Knot theory is an important branch of mathematics with applications in other branches of science. In this paper, we explore invariants on a special class of knots, known as virtual knots. We find new invariants by taking quotients of quandles, and introducing the fundamental Latin Alexander quandle and its Grobner basis. We also demonstrate examples of computations of these invariants.

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.

I Quandle sono strutture algebriche non associative e auto-distributive che emergono in maniera naturale in diverse aree della matematica. Le più importanti sono probabilmente la teoria dei nodi e lo studio delle soluzioni della (set-theoretical) Quantum Yang Baxter Equation, in relazione alla classificazione delle pointed Hopf Algebras. La varietà dei quandle rappresenta anche un ottimo esempio di varietà idempotente nel contesto dell’algebra universale. L’argomento della tesi é la teoria delle estensioni di quandle, ossia la descrizione delle proprietà di un quandle a partire dalle proprietà delle sue congruenze e dei suo quozienti. Nella tesi sono usate sia tecniche di algebra universale che di teoria delle categorie. Nei primi capitoli vengono presentati alcuni risultati noti sui quandle, e in alcuni casi questi vengono estesi anche ai Left-quasigroup. Nei capitoli successivi, prima vengono analizzate alcune classi definite tramite l’esistenza di specifiche operazioni (come la classe dei quandle Latini, Maltsev e Taylor), in seguito classi definite tramite l’esistenza di specifiche rappresentazioni (come i quandle omogenei, connessi, principali e affini) e infine le loro reciproche interazioni. Inoltre abbiamo approfondito la relazione tra il reticolo delle congruenze di un quandle e quello dei sottogruppi normali del suo gruppo delle trasvezioni. Il loro rapporto é molto stretto in virtù dell’autodistributività dei quandle. Infatti tra i due reticoli esiste una corrispondenza di Galois. Alcune proprietà delle congruenze, quali Abelianess, centrality e strongly Abelianness hanno una descrizione in termini di teoria dei gruppi. Lo stesso si può dire per altre proprietá come risolubilitá e nilpotenza (dal punto di vista dell’algebra universale). In particolare quest’ultima corrispondenza é più marcata per i quandle di tipo Taylor. La struttura delle estensioni di un quandle ha una descrizione combinatorica per mezzo della (non-Abelian) Cohomology. Alcune famiglie di estensioni vengono prese in esame più in dettaglio. I covering, i quali corrispondono alle congruenze di tipo strongly Abelian, e le estensioni Abeliane che invece corrispondono alle congruenze di tipo Abelian i cui blocchi sono connessi. Abbiamo definito una particolare famiglia di cocicli, detti normalizzati, i quali hanno reso possibile il calcolo esplicito della cohomologia per alcune classi di quandle Latini. Ad esempio i quandle connessi che ammettono una rappresentazione affine su gruppi ciclici e i quandle 2-transitivi hanno cohomologia banale. Infine, abbiamo presentato un approccio di tipo categoriale ai covering. Questo approccio, dovuto ad Eisermann, si basa sulle proprietà dell’Adjoint group di un quandle. Una certa rilevanza ha il rapporto tra covering e estensioni centrali di gruppi. Il capitolo si chiude con la descrizione della classe dei quandle semplicemente connessi, cioè i quandle che non ammettono covering connessi non banali.
Quandles are non associative and self-distributive algebraic structures which arise very naturally in a lot of different areas of mathemathics. The most important probably are knot theory and the study of the solution of the (set-theoretical) Quantum Yang Baxter Equation which is related to the classification of pointed Hopf Algebras. They also provide a good example of idempotent variety in universal algebra. The thesis is about extension theory of quandles, i.e., the description of the properties of a quandle starting from the properties of its congruences and the properties of its homomorphic images. We use both universal algebraic and categorical technique. In the first Sections, we have summarized some known results on quandles and we have extended some constructions to Left-quasigroups. In the following, first we analize some classes defined by existence of specific term operations (as Latin quandles, Maltsev quandles and Taylor quandles), then classes defined by existence of specific representations (as homogeneous, connected, principal and affine quandles) and their mutual relations. We also investigate the relation between the congruence lattice of a quandle and lattice of normal subgroups of its transvection group. This relation in tight by virtue of the self-distributivity property of quandles. We show that there exists a Galois connection between these two lattices. Universal algebraic properties of congruences as Abelianness, centrality and strongly Abelianness have a group-theoretical counterpart. The same is true for other universal algebraic notions as solvability and nilpotency (in particular for Taylor quandles). A combinatorial description of the structure of extensions of a given quandle, through (non-Abelian) Cohomology is presented. In particular we focus on the special family of coverings of a given quandle, with turns out to correspond to strongly Abelian congruences. We also define Abelian extensions which correspond to Abelian congruences with connected blocks. We also compute explicitly cohomology for some classes of Latin quandles, by using the properties of normalized cocycles. We show that connected affine quandles over cyclic group and doubly transitive quandles have trivial cohomology. Finally we present a categorical approach to coverings due to Eisermann, by using the Adjoint group of a quandle and the connection between coverings and central extensions of groups. The class of simply connected quandles, namely the class of connected quandles with no proper coverings, have been characterized.

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.

Quandles, which are algebraic structures related to knots, can be used to color knot diagrams, and the number of these colorings is called the quandle coloring invariant. We strengthen the quandle coloring invariant by considering a graph structure on the space of quandle colorings of a knot, and we call our graph the quandle coloring quiver. This structure is a categorification of the quandle coloring invariant. Then, we strengthen the quiver by decorating it with Boltzmann weights. Explicit examples of links that show that our enhancements are proper are provided, as well as background information in quandle theory.

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.

We study the difference between quandles that arise from conjugation in groups and those which do not. As a result, we define conjugation subquandles, and show that not all quandles or keis are in this class of examples. We investigate coloring by keis which are not conjugation subquandles. And we investigate the relationship between decomposable quandles or keis and link colorings. Subsequently, we analyze what kinds of quandles or keis can be homomorphic images of a knot quandle.

We present the theory of selfdistributive quasigroups and the construction of non-affine selfdistributive quasigroup of size 216 that was presented by Onoi in 1970 and which was the least known example of such structure of size 2k . Based on this construction, we introduce the notion of Onoi structures and Onoi mappings between them which generalizes Onoi's construction and which allows us to construct non-affine selfdistributive quasigroups of size 22k for k ≥ 3. We present and implement algorithm for finding central extensions of self- distributive quasigroups which enables us to classify non-affine selfdistributive quasigroups of size 2k and prove that those quasigroup exists exactly for k ≥ 6, k ̸= 7. We use this algorithm also in order to better understand the structure of non-affine selfdistributive quasigroups of size 26 . 1

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