Researcher ORCID Identifier
0009-0006-2867-1799
Graduation Year
2026
Document Type
Open Access Senior Thesis
Degree Name
Bachelor of Arts
Department
Mathematics
Reader 1
Sam Nelson
Reader 2
Chris Towse
Terms of Use & License Information
Abstract
Knot Theory is a vast and diverse subfield of modern mathematics involving the classification and abstraction of knots and links. In this thesis, we wish to provide the necessary background for and an explanation of two papers in different subfields of knot theory, The Forbidden Quiver of a Link, and Biquandle Fares and Link Invariants.
In Chapter I, we begin with an introduction to knot theory and knot invariants. We continue to present the example of Fox Colorings, and conclude the chapter with an example of the Fox Coloring Number Invariant.
In Chapter II, we explore the derivation and utilization of Gauss Diagrams in knot theory. We discuss Permitted and Forbidden moves. We then define and prove the existence of Virtual Knots.
In Chapter III, we introduce Forbidden moves on a Gauss Diagram and develop several invariants associated with the Forbidden Quiver of a Link.
In Chapter IV, we provide a brief introduction to Biquandles and Biquan- dle colorings of knots.
In Chapter V, we begin with an introduction to Biquandle Fares. We then construct three knot invariants utilizing Biquandle Fares: The complete fare, the Through fare, and the Crooked Fare.
We conclude with some directions for future research.
Recommended Citation
Shah, Stella, "An Introduction to Modern Conversations on Knot Invariants" (2026). Scripps Senior Theses. 2784.
https://scholarship.claremont.edu/scripps_theses/2784
