Quantum knot mosaics and the growth constant

Lomonaco and Kauffman introduced a knot mosaic system to give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This paper is inspired by an open question about the knot mosaic enumeration suggested by them. A knot $n$--mosaic is an $n \times n$ array of 11 mosaic tiles representing a knot or a link diagram by adjoining properly that is called suitably connected. The total number of knot $n$--mosaics is denoted by $D_n$ which is known to grow in a quadratic exponential rate. In this paper, we show the existence of the knot mosaic constant $\delta = \lim_{n \rightarrow \infty} D_n^{\ \frac{1}{n^2}}$ and prove that $$4 \leq \delta \leq \frac{5+ \sqrt{13}}{2} \ (\approx 4.303).$$