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Full-Text Articles in Physical Sciences and Mathematics
Non-Associative Algebraic Structures In Knot Theory, Emanuele Zappala
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
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
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 …