Department of Mathematics
University of Georgia

Summer REU: Knots, Satellites and Immersed Curves

Leaders: Akram Alishahi and Carolyn Engelhardt

Dates: June 1, 2026 - July 24, 2026

Description: A knot is essentially a tangled loop of string in 3D space. To study them, we often project them onto a 2D plane to create a knot diagram—a snapshot that captures every twist and crossover. However, these diagrams hide a deeper complexity: How can we prove two knots are the "same" if one can be wiggled into the shape of the other? How many times must a knot "pass through itself" for it to become unknotted? Or more abstractly, how does a knot behave when it is allowed to pass through itself in four-dimensional space?

This project moves beyond basic diagrams into the world of satellite knots. Imagine taking one knot (the "pattern") and tying it inside a solid tube, then tying that tube into the shape of a second knot (the "companion"). Specifically, we will focus on (1,1)-satellite patterns. While the name sounds technical, these are effectively knots that can be described by drawing a simple diagram on the surface of a torus (a doughnut shape). Because of this accessible structure, we can study them using immersed curve techniques. This modern approach allows us to translate difficult algebraic calculations into a visual geometry problem: we compute invariants simply by analyzing how curves wrap around and intersect on the surface of the torus.

Our primary objective is to compute invariants that help us understand if a knot can bound a smooth disk in 4D space or how many times must a knot pass through itself to become unknotted.

Support:

• Stipend: each student will receive a stipend of $4,800 ($600 per week)
• Housing: directly paid on-campus dorms (pending approval), or reimbursement up to $2,160
• Meals: each student will receive a total per diem of $2000 for their meals. This will cover the cost of breakfast, lunch and dinner at Bolton Dining Commons on campus.
• Travel: A partial travel support to Athens, GA, up to $600

Eligibility: Must be US Citizen or Permanent Resident.

Activities: Students will spend an intensive 8 weeks engaged in collaborative activities such as the following:

• Lectures and discussions: Students will learn the basic and modern tools for studying knots theory
• Problem sessions: Students will put their new skills to practice by working together to solve problems with known solutions.
• Research reports and presentations: Students will discuss their progress and improve their math communication skills.

Background / Prerequisites: Students are expected to have some experience with linear algebra, and to have an interest in studying unsolved problems in mathematics. While a formal course in proof-writing is not strictly required, the well-prepared participant should be willing to think carefully and deeply, and to be able to express their ideas clearly.

Application: Applications are accepted via mathprograms.org

Interested students should submit:

• Unofficial transcript
• A personal statement (no more than one page): Tell us about yourself: What topics and subjects are you interested in? What courses have you most enjoyed? Tell us why you are interested in participating in this REU, and how this experience would help you achieve your future goals. If you have any relevant research or work experience, or relevant skills, feel free to let us know. Finally, we expect our students to work together as a group. Briefly explain how you have worked well with others in the past, and how you would help the group work together smoothly.
• One letter of reference

Deadline: March 8, 2026

(Applications received after the deadline may still be considered. )

This REU is supported by the NSF CAREER grant DMS-2238103.