Graduation Year
2024
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
Idempotents, Knots, Quandle 2 Cocycle, Quandle rings, Quandles
Abstract
Quandles are sets with self-distributive binary operations that axiomatize the three Reidemeister movesin classical knot theory. In an attempt to bring ring theoretic techniques to the study of quandles, a theory of quandle rings analogous to the classical theory of group rings where several interconnections between quandles and their associated quandle rings have been explored. Functoriality of the construction implies that morphisms of quandle rings give a natural enhancement of the well-known quandle coloring and quandle 2 cocycle invariant of knots and links.
The dissertation is structured into two main parts. In the first part, we delve into quandle rings obtained from non-trivial quandles over rings. We demonstrate that integral quandle rings emerging from non-trivial involutory coverings possess infinitely many non-trivial idempotents which, themselves form quandles, contributing to a comprehensive understanding of their structure. Applying these findings to knot theory, we deduce that the quandle ring associated with the knot quandle of a non-trivial long knot exhibits non-trivial idempotents. Furthermore, we explore free products of quandles and establish that integral quandle rings of free quandles exclusively feature trivial idempotents, yielding an infinite family of such quandles.
In the second part, we focus on leveraging idempotents in quandle rings to enhance the quandle 2-cocycle invariant of knots and links. By combining idempotents with state sum invariants of knots, we successfully distinguish all 12965 prime oriented knots with up to 13 crossings, utilizing only 21 connected quandles and three quandles made of idempotents in quandle rings. Additionally, we distinguish from knots their mirror images using the same set of 24 quandles.
Scholar Commons Citation
Swain, Dipali, "Quandle Rings, Idempotents and Cocycle Invariants of Knots" (2024). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/10249
