Quantum knots and the number of knot mosaics

Abstract

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