Growth rate of quantum knot mosaics

Abstract

Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper ‘Quantum knots and mosaics’ to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. This paper is inspired by an open question about the knot mosaic enumeration suggested by them. A knot (m, n)-mosaic is an $m \times n$ array of 11 mosaic tiles representing a knot or a link diagram by adjoining properly. The total number $D_{m,n}$ of knot (m, n)-mosaics, which indicates the dimension of the Hilbert space of the quantum knot system, is known to grow in a quadratic exponential rate. Recently, the first author showed the existence of the knot mosaic constant $\delta = \lim _{m, n \rightarrow \infty } (D_{m,n})^{\frac{1}{mn}}$ and proved $4 \le \delta \le \frac{5+ \sqrt{13}}{2} = 4.302\cdots$ by developing an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. In this paper, we give a simpler proof of the lower bound and improve the upper bound of the knot mosaic constant as

$$\begin{aligned} 4 \le \delta \le 4.113\cdots \end{aligned}$$

by introducing two new concepts: quasimosaics and cling mosaics.