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Tabulating knot mosaics: crossing number 10 or less

Aaron Heap, Douglas Baldwin, James Canning and Greg Vinal

Vol. 18 (2025), No. 1, 91–104
Abstract

The study of knot mosaics is based upon representing knot diagrams using a set of tiles on a square grid. This branch of knot theory has many unanswered questions, especially regarding the efficiency with which we draw knots as mosaics. While any knot or link can be displayed as a mosaic, for most of them it is still unknown what size of mosaic (mosaic number) is necessary and how many nonblank tiles (tile number) are necessary to depict a given knot or link. We implement an algorithmic programming approach to find the mosaic number and tile number of all prime knots with crossing number 10 or less. We also introduce an online repository which includes a table of knot mosaics and a tool that allows users to create and identify their own knot mosaics.

Keywords
knot theory, knot mosaic, tile number, space efficient, mosaic number, table of knots
Mathematical Subject Classification
Primary: 57-08, 57K10
Supplementary material

Mosaics for the knots referenced in the theorems.

Milestones
Received: 23 March 2023
Revised: 25 May 2023
Accepted: 11 July 2023
Published: 24 January 2025

Communicated by Colin Adams
Authors
Aaron Heap
Department of Mathematics
State University of New York at Geneseo
Geneseo, NY
United States
Douglas Baldwin
Department of Mathematics
State University of New York at Geneseo
Geneseo, NY
United States
James Canning
State University of New York at Geneseo
Laurel, MD
United States
Greg Vinal
Department of Mathematics
University of Buffalo
Buffalo, NY
United States