An n-dimensional rep-tile is a PL n-manifold X, embedded in $\mathbb{R}^n$, which can be decomposed as the union of mutually isometric manifolds similar to X which have non-overlapping interiors. For one astonishing example, all knot exteriors are homeomorphic to rep-tiles, by a 2021 result of Blair, Marley and Richards. I will give an isotopy classification of rep-tiles in all dimensions. I will also outline our proof, which is based on a technique called ball swapping. This is joint work with Ryan Blair, Patricia Cahn and Hannah Schwartz.

https://arxiv.org/abs/2412.19986