Constructions of and Bounds on the Toric Mosaic Number

Kendall Heiney; Margaret Kipe; Samantha Pezzimenti; Kaelyn Pontes; Luc Ta

doi:10.1016/j.topol.2025.109657

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arXiv · 2504.02265

Constructions of and Bounds on the Toric Mosaic Number

Kendall Heiney·Margaret Kipe·Samantha Pezzimenti·Kaelyn Pontes·Luc Ta

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Abstract

Knot mosaics were introduced by Kauffman and Lomonaco in the context of quantum knots, but have since been studied for their own right. A classical knot mosaic is formed on a square grid. In this work, we identify opposite edges of the square to form mosaics on the surface of a torus. We provide two algorithms for efficiently constructing toric mosaics of torus knots, providing upper bounds for the toric mosaic number. Using these results and a computer search, we provide a census of known toric mosaic numbers.

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Kendall Heiney, Margaret Kipe, Samantha Pezzimenti, Kaelyn Pontes, Luc Ta. 2025-04-03. Constructions of and Bounds on the Toric Mosaic Number. https://doi.org/10.1016/j.topol.2025.109657

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