Knot Theory: Exploring the Relationship Between the Alexander Polynomial and Alexander Tribracket Knot Invariants
Campus Only Senior Thesis
Bachelor of Arts
Knot theory is the study of tangled strings whose ends are attached to form a closed loop. The primary question that knot theory seeks to answer is how can we determine if two knots are `equivalent', or in other words, can be untangled into each other? A knot invariant is a quantity corresponding to a knot that acts like a fingerprint and helps us distinguish when two knots are different. Early knot invariants came in the form of booleans (true/false) and integers, but within the last century knot invariants in the form of polynomials and algebraic structures have been discovered. In this thesis we make progress on an open question in knot theory: Can we derive the Alexander polynomial knot invariant from the Alexander tribracket, a knot invariant in the form of an algebraic structure? We prove that we can derive the Alexander polynomial from the Alexander tribracket for a certain class of knots and links and conjecture a formula for deriving the Alexander polynomial from the Alexander tribracket for all knots and links.
Wartman, Lily Natasha, "Knot Theory: Exploring the Relationship Between the Alexander Polynomial and Alexander Tribracket Knot Invariants" (2023). Scripps Senior Theses. 2113.
This thesis is restricted to the Claremont Colleges current faculty, students, and staff.