Quantum knot mosaics and bounds of the growth constant

The discovery of the Jones polynomial made an important connection between quantum physics and knot theory. Kauffman and Lomonaco introduced the knot mosaic system to define the quantum knot system for the purpose of representing 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 × n array of 11 mosaic tiles representing a knot diagram by adjoining properly. The total number Dm×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, Oh et al. developed the state matrix recursion method producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. Furthermore, they showed the existence of the knot mosaic constant =limm,n→∞(Dm×n) 1 mn and found its upper and lower bounds in a series of papers. The latest upper bound was obtained through two new concepts: quasimosaics and cling mosaics. As a sequel to this research program, we adjust the state matrix recursion method to handle cling mosaics inside a quasimosaic, which is called the progressive state matrix recursion method. This method provides recursive matrix-relations producing a sharper bound of the knot mosaic constant: 4 ≤ δ ≤ 4.1035507⋯.