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

TL;DR

This paper investigates the structure of three-dimensional domino tilings of cylinders over certain disks, establishing conditions under which all tilings with the same twist are connected via flips, thus demonstrating regularity.

Contribution

It proves that hamiltonian disks with narrow bottlenecks are regular, extending understanding of tiling connectivity and invariants in 3D domino tilings.

Findings

Hamiltonian disks with narrow bottlenecks are regular.

Absence of bottlenecks implies regularity.

Connectivity of tilings with same twist is achieved through flips.

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.