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Cubic knot
mosaics
Samantha Pezzimenti, Trevor Meintel and Aaron Shabon |
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Vol. 18 (2025), No. 4, 629–642
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Abstract |
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Knot projections can be constructed from a set of specific mosaic tiles, which were first introduced by Lomonaco and Kauffman in 2008. Typically, the tiles are arranged in a square grid and invariants such as the mosaic number (minimal size grid needed to build the knot with tiles) can be defined. We extend this idea by constructing knot mosaics on the surface of a three-dimensional object — in particular, a cube. In this paper, we introduce the notion of cubic knot mosaics and define the cubic mosaic number of a knot. We compute the cubic mosaic numbers for prime knots up to 8 crossings and explore some bounds on the relationship between crossing number and cubic mosaic number. Finally, we provide some open questions and possible areas to explore. |
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Keywords
knots, knot mosaics, knot theory, mosaic number, cubic
mosaics, cubic knot mosaics, mosaic invariant
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Mathematical Subject Classification
Primary: 57K10
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Milestones
Received: 21 June 2023
Revised: 7 April 2024
Accepted: 16 April 2024
Published: 30 July 2025
Communicated by Colin Adams
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Authors |
| Samantha Pezzimenti | |
| Department of Mathematics Penn State Brandywine Media, PA United States |
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| Trevor Meintel | |
| Pennsylvania State University University Park, PA United States |
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| Aaron Shabon | |
| Pennsylvania State Univsersity University Park, PA United States |
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| © 2025 MSP (Mathematical Sciences Publishers). |
