Physical Sciences and Mathematics Commons^™

Non-Associative Algebraic Structures In Knot Theory, Emanuele Zappala

USF Tampa Graduate Theses and Dissertations

In this dissertation we investigate self-distributive algebraic structures and their cohomologies, and study their relation to topological problems in knot theory. Self-distributivity is known to be a set-theoretic version of the Yang-Baxter equation (corresponding to Reidemeister move III) and is therefore suitable for producing invariants of knots and knotted surfaces. We explore three different instances of this situation. The main results of this dissertation can be, very concisely, described as follows. We introduce a cohomology theory of topological quandles and determine a class of topological quandles for which the cohomology can be computed, at least in principle, by means of …

Generalizations Of Quandles And Their Cohomologies, Matthew J. Green

USF Tampa Graduate Theses and Dissertations

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 …

Contributions To Quandle Theory: A Study Of F-Quandles, Extensions, And Cohomology, Indu Rasika U. Churchill

USF Tampa Graduate Theses and Dissertations

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 …

Quandle Coloring And Cocycle Invariants Of Composite Knots And Abelian Extensions, W Edwin Clark, Masahico Saito, Leandro Vendramin

Mathematics and Statistics Faculty Publications

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle coloring of the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of …