Tile Numbers of Knot Corner Mosaics
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910930481184768 |
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| author | Aylaian, Ezra |
| author_facet | Aylaian, Ezra |
| contents | A knot mosaic is a grid of pictorial tiles representing a tame knot or link. Recently, two groups independently introduced a new set of tiles. We call mosaics made with these new tiles corner mosaics. The (corner) tile number is the minimum number of tiles needed to represent a knot or link as a (corner) mosaic. We show that the corner tile number lies strictly between the tile number and $3$ times the tile number, resolving a question of Heap et al. We also show that the only knots and links with corner tile number $<12$ and no unlinked, unknotted components are the Hopf link $P(1,1)$, the trefoil knot $3_1$, Solomon's knot $P(1,1,1,1)$, the connect sum of two Hopf links $P(1,1) \# P(1,1)$, the cinquefoil knot $5_1$, the star of David link $P(1,1,1,1,1,1)$, the figure-eight knot $4_1$, and the three-twist knot $5_2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_12258 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Tile Numbers of Knot Corner Mosaics Aylaian, Ezra Geometric Topology 57K10 (Primary) 05B30, 05B50 (Secondary) A knot mosaic is a grid of pictorial tiles representing a tame knot or link. Recently, two groups independently introduced a new set of tiles. We call mosaics made with these new tiles corner mosaics. The (corner) tile number is the minimum number of tiles needed to represent a knot or link as a (corner) mosaic. We show that the corner tile number lies strictly between the tile number and $3$ times the tile number, resolving a question of Heap et al. We also show that the only knots and links with corner tile number $<12$ and no unlinked, unknotted components are the Hopf link $P(1,1)$, the trefoil knot $3_1$, Solomon's knot $P(1,1,1,1)$, the connect sum of two Hopf links $P(1,1) \# P(1,1)$, the cinquefoil knot $5_1$, the star of David link $P(1,1,1,1,1,1)$, the figure-eight knot $4_1$, and the three-twist knot $5_2$. |
| title | Tile Numbers of Knot Corner Mosaics |
| topic | Geometric Topology 57K10 (Primary) 05B30, 05B50 (Secondary) |
| url | https://arxiv.org/abs/2311.12258 |