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Tiling the 4-ball with knotted surfaces

James Ross, Hannah Schwartz, Andrew Ye

TL;DR

This work solves a 4D analogue of tiling a cube by congruent knotted objects: for any closed orientable surface $F$ in $\mathbb{R}^4$ and any $n \ge 3$, the unit $4$-ball $B^4$ can be monohedrally tiled by $n$ tiles each congruent to a regular neighborhood of $F$ (with corners). The method combines bridge trisections of Meier–Zupan with a Complement Covering Theorem to decompose $B^4$ into a neighborhood of $F$ and a pair of $4$-balls, then generalizes Oh's 3D tiling construction to four dimensions via a pinwheel-like assembly to obtain arbitrary $n \ge 3$. The results answer Adams's question on higher-dimensional analogues of tilings by knotted tori, extend the 3D tiling framework to $4$-dimensions, and provide a structured approach to tilings by thickened knotted surfaces in $B^4$. This establishes a robust foundation for high-dimensional rep-tiles and tilings involving knotted surfaces, with potential implications for isotopy classifications and geometric decompositions in 4-manifold topology.

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.

Tiling the 4-ball with knotted surfaces

TL;DR

This work solves a 4D analogue of tiling a cube by congruent knotted objects: for any closed orientable surface in and any , the unit -ball can be monohedrally tiled by tiles each congruent to a regular neighborhood of (with corners). The method combines bridge trisections of Meier–Zupan with a Complement Covering Theorem to decompose into a neighborhood of and a pair of -balls, then generalizes Oh's 3D tiling construction to four dimensions via a pinwheel-like assembly to obtain arbitrary . The results answer Adams's question on higher-dimensional analogues of tilings by knotted tori, extend the 3D tiling framework to -dimensions, and provide a structured approach to tilings by thickened knotted surfaces in . This establishes a robust foundation for high-dimensional rep-tiles and tilings involving knotted surfaces, with potential implications for isotopy classifications and geometric decompositions in 4-manifold topology.

Abstract

We show that for any closed, orientable surface smoothly embedded in , the unit -ball can be tiled using tiles each congruent to a regular neighborhood (with corners) of a surface smoothly isotopic to . This gives a 4-dimensional analog of tilings of the -ball that were constructed in the 90's using congruent knotted tori.
Paper Structure (4 sections, 4 theorems, 1 equation, 16 figures)

This paper contains 4 sections, 4 theorems, 1 equation, 16 figures.

Table of Contents

  1. Introduction
  2. Tilings, Tangles, and Trisections
  3. Tiling the 3-ball with trivial tangles
  4. Tiling the 4-ball with knotted surfaces

Key Result

Theorem 1

Let $F$ be any closed, orientable surface smoothly embedded in $\mathbb{R}^4$ considered up to smooth isotopy. For any $n\geq 3$, the unit $4$-ball in $\mathbb{R}^4$ admits a monohedral tiling with $n$ tiles, each of which is congruent in $\mathbb{R}^4$ to a neighborhood of the surface $F$.

Figures (16)

  • Figure 1: A monohedral tiling, see Definition \ref{['monohedraltile']}, of a rectangle in $\mathbb{R}^2$, where all tiles are related by an isometry of the plane (rotation, translation, reflection).
  • Figure 2: An example of a trivial tangle with two arcs and four strands.
  • Figure 3: A tangle stabilized twice, where section (a) is the top section, section (b) is a crossing section, and section (c) is a trivial section.
  • Figure 4: A PL depiction (drawn down one dimension) of the $0$-trisection of a $4$-ball into three separate $4$-balls $P_{ij}$, whose triple intersection is the binding $\Sigma$
  • Figure 5: Top down view of a trivial section.
  • ...and 11 more figures

Theorems & Definitions (12)

  • Theorem 1: 4D Tiling Theorem
  • Theorem 2: Complement Covering Theorem
  • Theorem 3: 3D Tiling Theorem
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • ...and 2 more