# ALGEBRAIC COMBINATORICS

Domino tilings and flips in dimensions 4 and higher
Algebraic Combinatorics, Volume 5 (2022) no. 1, pp. 163-185.

In this paper we consider domino tilings of bounded regions in dimension $n\ge 4$. We define the twist of such a tiling, an elements of $ℤ/\left(2\right)$, and prove that it is invariant under flips, a simple local move in the space of tilings.

We investigate which regions $𝒟$ are regular, i.e. whenever two tilings ${\mathbf{t}}_{0}$ and ${\mathbf{t}}_{1}$ of $𝒟×\left[0,N\right]$ have the same twist then ${\mathbf{t}}_{0}$ and ${\mathbf{t}}_{1}$ can be joined by a sequence of flips provided some extra vertical space is allowed. We prove that all boxes are regular except $𝒟={\left[0,2\right]}^{3}$.

Furthermore, given a regular region $𝒟$, we show that there exists a value $M$ (depending only on $𝒟$) such that if ${\mathbf{t}}_{0}$ and ${\mathbf{t}}_{1}$ are tilings of equal twist of $𝒟×\left[0,N\right]$ then the corresponding tilings can be joined by a finite sequence of flips in $𝒟×\left[0,N+M\right]$. As a corollary we deduce that, for regular $𝒟$ and large $N$, the set of tilings of $𝒟×\left[0,N\right]$ has two twin giant components under flips, one for each value of the twist.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.205
Classification: 05B45,  52C20,  52C22,  05C70
Keywords: Higher dimensional tilings, dominoes, dimers
Klivans, Caroline J. 1; Saldanha, Nicolau C. 2

1 Division of Applied Mathematics, Box F 182 George Street Brown University Providence, RI 02912 USA
2 Departamento de Matemática, PUC-Rio Rua Marquês de São Vicente 225, Rio de Janeiro, RJ 22451-900 Brazil
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Klivans, Caroline J.; Saldanha, Nicolau C. Domino tilings and flips in dimensions 4 and higher. Algebraic Combinatorics, Volume 5 (2022) no. 1, pp. 163-185. doi : 10.5802/alco.205. https://alco.centre-mersenne.org/articles/10.5802/alco.205/

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