Cubic knot mosaics

Pezzimenti, Samantha; Meintel, Trevor; Shabon, Aaron

doi:10.2140/involve.2025.18.629

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Cubic knot mosaics

Samantha Pezzimenti, Trevor Meintel and Aaron Shabon

Vol. 18 (2025), No. 4, 629–642

DOI: 10.2140/involve.2025.18.629

Abstract

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.

Keywords

knots, knot mosaics, knot theory, mosaic number, cubic mosaics, cubic knot mosaics, mosaic invariant

Mathematical Subject Classification

Primary: 57K10

Milestones

Received: 21 June 2023

Revised: 7 April 2024

Accepted: 16 April 2024

Published: 30 July 2025

Communicated by Colin Adams

Authors

Samantha Pezzimenti
Department of Mathematics Penn State Brandywine Media, PA United States

Trevor Meintel
Pennsylvania State University University Park, PA United States

Aaron Shabon
Pennsylvania State Univsersity University Park, PA United States

Vol. 18, No. 4, 2025