# Period and toroidal knot mosaics

## TL;DR

This paper introduces period and toroidal knot mosaics, providing algorithms for their enumeration and analyzing their growth rates, thereby extending knot mosaic theory with new variants relevant in physics and mathematics.

## Contribution

It presents novel variants of knot mosaics—periodic and toroidal—and algorithms for their exact enumeration, along with asymptotic analysis of their growth rates.

## Key findings

- Exact enumeration algorithms for period knot mosaics for all positive integers m,n.
- Enumeration of toroidal knot mosaics for co-prime and prime p cases.
- Analysis of asymptotic growth rates of mosaic cardinalities.

## Abstract

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on `Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an $m \! \times \! n$ matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot (m,n)-mosaics for any positive integers m and n, toroidal knot (m,n)-mosaics for co-prime integers m and n, and furthermore toroidal knot (p,p)-mosaics for a prime number p. We also analyze the asymptotics of the growth rates of their cardinality.