Knot mosaic tabulation

TL;DR

This paper explores the mosaic number of knots within the knot mosaic system, determining it for all prime knots with up to eight crossings, and aims to make the topic accessible to undergraduates.

Contribution

It provides the first comprehensive determination of the mosaic number for all prime knots with up to eight crossings, simplifying the concept for educational purposes.

Findings

Mosaic number determined for all prime knots with ≤8 crossings

Introduces an accessible approach to knot mosaic theory for undergraduates

Establishes the mosaic number as a fundamental knot invariant

Abstract

In 2008, Lomonaco and Kauffman introduced a knot mosaic system to define a quantum knot system. A quantum knot is used to describe a physical quantum system such as the topology or status of vortexing that occurs on a small scale can not see. Kuriya and Shehab proved that knot mosaic type is a complete invariant of tame knots. In this article, we consider the mosaic number of a knot which is a natural and fundamental knot invariant defined in the knot mosaic system. We determine the mosaic number for all eight-crossing or fewer prime knots. This work is written at an introductory level to encourage other undergraduates to understand and explore this topic. No prior of knot theory is assumed or required.