Domino tilings of cylinders: the domino group and connected components under flips

TL;DR

This paper studies the structure of domino tilings in three-dimensional regions, introducing the domino group and characterizing when tilings can be connected via flips based on the disk's properties.

Contribution

It defines the domino group for quadriculated disks, characterizes regular disks via this group, and explores the structure of tilings and flips in three-dimensional regions.

Findings

Regular disks have domino groups isomorphic to Z ⊕ Z/(2).

For rectangles with even area, regularity depends on side lengths being at least 3.

Non-regular disks have non-abelian domino groups with exponential growth.

Abstract

We consider domino tilings of three-dimensional cubiculated regions. A flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is an integer associated to each tiling, which is invariant under flips. A balanced quadriculated disk $D$ is regular if whenever two tilings $t_0$ and $t_1$ of $D \times [0,N]$ have the same twist then $t_0$ and $t_1$ can be joined by a sequence of flips provided some extra vertical space is allowed. We define the domino group of a quadriculated disk and prove that $D$ is regular if and only if its domino group is isomorphic to $Z \oplus Z/(2)$. We prove that a rectangle $D = [0,L] \times [0,M]$ with $LM$ even is regular if and only if $\min\{L,M\} \ge 3$ and conjecture that in general "large" disks are regular. In the cases where $D$ is not regular we prove partial results concerning the structure of the domino group: the group is not abelian and has exponential growth. We also prove that if $D$ is regular then the extra vertical space necessary to join by flips two tilings of $D \times [0,N]$ with the same twist depends only on $D$, not on the height $N$.