Quantum knots and the number of knot mosaics

Lomonaco and Kauffman developed a knot mosaic system to introduce a precise and workable definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot $$(m,n)$$(m,n)-mosaic is an $$m \times n$$m×n matrix of mosaic tiles ($$T_0$$T0 through $$T_{10}$$T10 depicted in the introduction) representing a knot or a link by adjoining properly that is called suitably connected. $$D^{(m,n)}$$D(m,n) is the total number of all knot $$(m,n)$$(m,n)-mosaics. This value indicates the dimension of the Hilbert space of these quantum knot system. $$D^{(m,n)}$$D(m,n) is already found for $$m,n \le 6$$m,n≤6 by the authors. In this paper, we construct an algorithm producing the precise value of $$D^{(m,n)}$$D(m,n) for $$m,n \ge 2$$m,n?2 that uses recurrence relations of state matrices that turn out to be remarkably efficient to count knot mosaics. $$\begin{aligned} D^{(m,n)} = 2 \, \Vert (X_{m-2}+O_{m-2})^{n-2} \Vert \end{aligned}$$D(m,n)=2?(Xm-2+Om-2)n-2?where $$2^{m-2} \times 2^{m-2}$$2m-2×2m-2 matrices $$X_{m-2}$$Xm-2 and $$O_{m-2}$$Om-2 are defined by $$\begin{aligned} X_{k+1} = \begin{bmatrix} X_k&O_k \\ O_k&X_k \end{bmatrix} \ \hbox {and } \ O_{k+1} = \begin{bmatrix} O_k&X_k \\ X_k&4 \, O_k \end{bmatrix} \end{aligned}$$Xk+1=XkOkOkXkandOk+1=OkXkXk4Okfor $$k=0,1, \cdots , m-3$$k=0,1,?,m-3, with $$1 \times 1$$1×1 matrices $$X_0 = \begin{bmatrix} 1 \end{bmatrix}$$X0=1 and $$O_0 = \begin{bmatrix} 1 \end{bmatrix}$$O0=1. Here $$\Vert N\Vert $$?N? denotes the sum of all entries of a matrix $$N$$N. For $$n=2$$n=2, $$(X_{m-2}+O_{m-2})^0$$(Xm-2+Om-2)0 means the identity matrix of size $$2^{m-2} \times 2^{m-2}$$2m-2×2m-2.