Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.21700 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914057382002688 |
|---|---|
| author | Foster, Leigh |
| author_facet | Foster, Leigh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21700 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stones, Bones, and Snakes: Tilability of the hexagonal grid via the double dimer model Foster, Leigh Combinatorics 05B45 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. |
| title | Stones, Bones, and Snakes: Tilability of the hexagonal grid via the double dimer model |
| topic | Combinatorics 05B45 |
| url | https://arxiv.org/abs/2509.21700 |