Domino tilings of cylinders: the domino group and connected components under flips
Abstract
We consider domino tilings of threedimensional 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 $N$.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 DOI:
 10.48550/arXiv.1912.12102
 arXiv:
 arXiv:1912.12102
 Bibcode:
 2019arXiv191212102S
 Keywords:

 Mathematics  Combinatorics;
 05B45
 EPrint:
 38 pages, 29 figures. Revised to clarify several passages