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×nn×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 DnDn which is known to grow in a quadratic exponential rate. In this paper, we show the existence of the knot mosaic constant δ=limn→∞⁡Dn1n2 and prove that4≤δ≤5+132(≈4.303).