Graduation Year
2025
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.
Committee Member
Dmytro Savchuk, Ph.D.
Committee Member
Boris Shekhtman, Ph.D.
Committee Member
Nasir Ghani, Ph.D.
Keywords
Torus knots, Quandle coloring quivers, polynomials, stuquandles
Abstract
This dissertation explores the algebraic and combinatorial structures arising from the study of knot theory through the lens of quandles, their colorings, and associated quiver and ring constructions. It is organized around three central themes: the analysis of quandle coloring quivers for torus links, the development of polynomial invariants and quiver invariants for stuck knots and links, and the algebraic and graphical study of quandle rings. The first chapter begins with an introduction to relevant concepts throughout the dissertation.
In the second chapter, we classify quandle coloring quivers of torus links using dihedral quandles. We describe these quivers as weighted complete digraphs, decomposed into disjoint components corresponding to coloring equivalence classes. Explicit formulas for edge weights are derived from quandle endomorphism actions, leading to a combinatorial understanding of link colorings enhanced by quiver structure.
The third and fourth chapters introduce stuck knots and links, which can be described as immersed curves with fixed crossing orientations inspired by physical and biological systems such as RNA folding. We define stuquandles as the algebraic structures governing these objects and construct new invariants by generalizing the quandle polynomial to subquandle and substuquandle polynomials and constructing the quandle coloring quiver invariant using stuquandles. These refined invariants successfully distinguish between stuck links and RNA arc diagrams not separated by traditional counting invariants.
The final chapter investigates quandle rings and their ring-theoretic and graph-theoretic properties. We prove that quandle rings over nontrivial quandles are never power-associative in characteristic zero and compute automorphism groups for quandle rings over various finite quandles. We further construct associated graphs, including zero-divisor graphs and automorphism graphs, revealing rich interactions between algebraic and combinatorial structures.
Together, this work contributes new invariants, frameworks, and classifications that enrich the use of quandles and quivers in knot theory, while also bridging connections to stuck knot theory, ring theory, and applications in biomathematics.
Scholar Commons Citation
Jones, Brooke, "Topics in Knot, Quandle, and Quiver Theory" (2025). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/10966
