Prime Theta-Curves from Torus Knots
Location
PANEL: Quantitative Approaches to Performance & Modeling
Science Center A254
Moderator: Abby Aresty
Document Type
Presentation - Open Access
Start Date
4-25-2025 3:00 PM
End Date
4-25-2025 4:00 PM
Abstract
Classical knot theory is the placement problem for a circle in Euclidean 3-space or in the 3-sphere. That problem has been generalized in numerous ways. Knotted graph theory replaces the circle with a graph. Knotting of certain graphs has been well-studied. That includes the theta-graph consisting of two vertices connected by three edges---a copy of the letter theta. A theta-curve is an embedding of the theta-graph in 3-space or in the 3-sphere. Connected sum of knots yields the notion of a prime knot. Similarly, one may sum two theta-curves at a vertex and one may sum a theta-curve and a knot along an edge. Those two operations yield the notion of a prime theta-curve. Our work was prompted by the question: How can one add an arc to a prime knot to obtain a prime theta-curve? We will show a way of doing that successfully.
Keywords:
Knot theory, Mathematics, Topology, Torus
Recommended Citation
Nieman, Sam and Calcut, Jack, "Prime Theta-Curves from Torus Knots" (2025). Research Symposium. 14.
https://digitalcommons.oberlin.edu/researchsymp/2025/presentations/14
Major
Mathematics
Project Mentor(s)
Jack Calcut, Mathematics
2025
Prime Theta-Curves from Torus Knots
PANEL: Quantitative Approaches to Performance & Modeling
Science Center A254
Moderator: Abby Aresty
Classical knot theory is the placement problem for a circle in Euclidean 3-space or in the 3-sphere. That problem has been generalized in numerous ways. Knotted graph theory replaces the circle with a graph. Knotting of certain graphs has been well-studied. That includes the theta-graph consisting of two vertices connected by three edges---a copy of the letter theta. A theta-curve is an embedding of the theta-graph in 3-space or in the 3-sphere. Connected sum of knots yields the notion of a prime knot. Similarly, one may sum two theta-curves at a vertex and one may sum a theta-curve and a knot along an edge. Those two operations yield the notion of a prime theta-curve. Our work was prompted by the question: How can one add an arc to a prime knot to obtain a prime theta-curve? We will show a way of doing that successfully.
