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

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.