Graduation Year

2017

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Mohamed Elhamdadi, Ph.D.

Co-Major Professor

Abdenacer Makhlouf, Ph.D.

Committee Member

Xiang-dong Hou, Ph.D.

Committee Member

Masahiko Saito, Ph.D.

Committee Member

David Rabson, Ph.D.

Keywords

Quandle, f-Quandle, Extension, Cohomology, Cocycle

Abstract

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.

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