Tiling the 4-ball with knotted surfaces

TL;DR

This paper proves that any closed, orientable surface embedded in four-dimensional space can be used to tile the four-dimensional ball with at least three congruent tiles, extending previous three-dimensional tiling results.

Contribution

It introduces a method to tile the 4-ball with congruent neighborhoods of knotted surfaces, generalizing 3D tiling constructions to four dimensions.

Findings

Any closed, orientable surface in R^4 can tile the 4-ball with at least 3 congruent pieces.

The tiles are neighborhoods of surfaces isotopic to the given surface.

This extends 3D knotted torus tilings to 4D surfaces.

Abstract

We show that for any closed, orientable surface $K$ smoothly embedded in $\mathbb{R}^4$, the unit $4$-ball $B^4 \subset \mathbb{R}^4$ can be tiled using $n \geq 3$ tiles each congruent to a regular neighborhood (with corners) of a surface smoothly isotopic to $K$. This gives a 4-dimensional analog of tilings of the $3$-ball that were constructed in the 90's using congruent knotted tori.