Project III 2024-25

Skateboard Tricks and Topological Flips

Martin Kerin

Photo by Daniel Lincoln on Unsplash

Description Suppose that A and B are two different skateboard flip tricks. Is it possible for a skater to perform a sequence of many tricks, such that consecutive tricks are very similar to each other and such that the first trick in the sequence is A while the last trick is B? One way to model this problem mathematically is to consider each flip trick as a continuous curve in the special orthogonal group SO(3) and to ask whether it is possible to continuously deform one into the other, a topological process known as homotopy. This project will study a recent preprint on this topic, which establishes the surprising result that, up to continuous deformations, there are only four flip tricks. Along the way, we will establish some necessary background on quaternions, topology and covering spaces. Furthermore, if time permits, we will investigate whether the results of this work can be generalised. Prerequisites Essential: Algebra II Topology II Corequisites Essential: Topology III Recommended: At least one of Geometry III Differential Geometry III Resources Justus Carlisle, Kyle Hammer, Robert Hingtgen and Gabriel Martins, Skateboard tricks and topological flips, preprint 2021

email: M Kerin