Tabulating Knot Mosaics: Crossing Number 10 or Less

TL;DR

The paper addresses tabulating knot mosaics for all prime knots with crossing number $c\le 10$. It introduces an automated pipeline that generates layouts, converts mosaics to Dowker–Thistlethwaite codes, reduces them, and identifies the knots, enabling exhaustive enumeration. The main results establish that every prime knot with $c\le 10$ has mosaic number $m\le 7$ and tile number $t\le 31$, and it provides a complete catalog along with mosaics from key theorems. An online repository, Knot Mosaic Space, accompanies the work, offering code, data, and interactive tools for constructing and identifying knot mosaics, thereby supporting further research and exploration in knot mosaic tabulation.

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 non-blank 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 can create and identify their own knot mosaics.

Figures (301)

• Figure 1: Possible tiles used for constructing knot mosaics.
• Figure 2: Examples of knot mosaics.
• Figure 3: All Space-efficient 6-mosaics, with number of non-blank tiles indicated
• Figure 4: All Space-efficient 7-mosaics, with number of non-blank tiles indicated
• Figure 6: Writing a knot mosaic as a matrix.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Corollary 8
• Conjecture