Domino tilings of three-dimensional cylinders: regularity of hamiltonian disks

TL;DR

This work addresses the connectivity of three-dimensional domino tilings in cylinders via local flips, using the twist invariant to classify tilings by a lattice-valued invariant. It develops the domino group framework and leverages Hamiltonian disks to reduce generators through flux considerations, proving regularity for bottleneck-free disks and for disks with narrow bottlenecks. A central outcome is that, for regular disks, the even domino group $G_D^+$ is cyclic and the twist $Tw: G_D^+ \to Z$ is an isomorphism, yielding $G_D \cong Z \oplus Z/2$. The results advance understanding of flip connectivity in 3D tilings and clarify how geometric features like bottlenecks control regularity and component structure.

Abstract

We consider three-dimensional domino tilings of cylinders $\mathcal{D} \times [0,N] \subset \mathbb{R}^3$, where $\mathcal{D} \subset \mathbb{R}^2$ is a balanced quadriculated disk and $N \in \mathbb{N}$. A flip is a local move in the space of tilings: two adjacent and parallel dominoes are removed and then placed in a different position. The twist is a flip invariant that associates an integer number to a domino tiling. A disk $\mathcal{D}$ is called regular if any two tilings of $\mathcal{D} \times [0,N]$ sharing the same twist can be connected through a sequence of flips once extra vertical space is added to the cylinder. We prove that hamiltonian disks with narrow and small bottlenecks are regular. In particular, we show that the absence of a bottleneck in a hamiltonian disk implies regularity.

Figures (17)

• Figure 1: A domino tiling of the cylinder $[0,4]\times [0,3] \times [0,4]$.
• Figure 2: Examples of disks whose regularity follows from Theorem \ref{['thm:regdisks']}.
• Figure 3: Examples of disks whose regularity follows from Theorem \ref{['thm:bottleneck']}.
• Figure 4: Four regular disks.
• Figure 5: The first and the second example show a hamiltonian cycle and a hamiltonian path in $[0,4]^2$. The third disk is path-hamiltonian but not hamiltonian. The fourth disk is neither path-hamiltonian nor hamiltonian.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof