Tiling the 4-ball with knotted 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 $n\ge 3$ 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.

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