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arXiv · 2509.21700

Stones, Bones, and Snakes: Tilability of the hexagonal grid via the double dimer model

Leigh Foster
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Abstract

The question of whether a given region can be successfully filled by a finite set of tiles has been commonly studied, and there are many available arguments for whether a given finite region can be tiled. We can show that there is no domino tiling of the mutilated chessboard via a coloring argument, and a slightly more subtle argument for other two-colored square-grid regions using a height function of Thurston. In this paper, we examine finite regions of the hexagonal grid and a set of tiles known as the stone, bone, and snake. Using matrices in $\text{SL}_2(\mathbb{C})$, we exhibit a new necessary criterion for a region to have a signed tiling by these tiles. This originally arose in a study of the double dimer model.

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Leigh Foster. 2025-09-25. Stones, Bones, and Snakes: Tilability of the hexagonal grid via the double dimer model. https://arxiv.org/abs/2509.21700

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