Domino tilings of cylinders: the domino group and connected components under flips
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 $N$.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2019
- DOI:
- 10.48550/arXiv.1912.12102
- arXiv:
- arXiv:1912.12102
- Bibcode:
- 2019arXiv191212102S
- Keywords:
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- Mathematics - Combinatorics;
- 05B45
- E-Print:
- 38 pages, 29 figures. Revised to clarify several passages