The easier restricted-tile conjecture for hexagonal mosaics

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For a knot, let a hexagonal rr-mosaic be a knot diagram assembled on the hexagonal rr-mosaic arrangement. The interior tiles are the tiles away from the boundary. The four three-crossing tiles are tiles 23, 24, 25, and 26, and the two three-arc tiles with no crossings are tiles 16 and 17. Easier conjecture. Every knot can be created as a hexagonal rr-mosaic for some rr, while restricting the interior tiles to these four three-crossing tiles and these two three-arc, crossing-free tiles. The source motivates this restriction using triple-crossing projections and presents it as an open computationally oriented strengthening of the general mosaic realization result.

References

Primary source

Hugh Howards, Jiong Li, Xiaotian Liu and Anna Paulec, “An infinite family of knots whose hexagonal mosaic number is only realized in non-reduced projections”, arXiv:1912.03697 (2026).

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