Linked Knots from the Gyro Operation on the Dodecahedron
Henriette Lipschütz, Martin Skrodzki, Ulrich Reitebuch, and Konrad Polthier

Proceedings of Bridges 2022: Mathematics, Art, Music, Architecture, Culture
Pages 175–182
Regular Papers

Abstract

John Horton Conway’s taxonomy for Archimedean and Catalan solids includes the snub and the dual operation, their combination being known as the gyro operation. Intriguingly, recent work showed that the gyro operation can be used to create weaving patterns when applied to planar n-gons. This process can be applied to two-dimensional polyhedral surfaces in ℝ3. In this paper, we investigate the weaving structures that are created on the dodecahedron. We show that the weaving structures can be interpreted via knot theory and written in form of a link diagram. Furthermore, we prove the number of knots arising on the dodecahedron to be six. Finally, we present hand-crafted physical representations of the results in form of balls made of yarn, inspired by Japanese temari.

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