A simple closed space curve that is comprised of quarter circles of fixed curvature with continuous tangents is known as a Tangle, after the popular fidget toy. We show that all Tangles of a given length n (or n-Tangles) correspond to the solutions of a particular system of polynomial equations. Using the software platform Macaulay2, we prove the nonexistence of 5-Tangles and describe all 6-Tangles.

Joint work with undergraduate student Katie Prowell.

Slides available at https://d-torrance.github.io/maa-se-2026-tangles/.