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)-mosaic is an $m \times n$matrixofmosaictiles($T_0$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. $$D^{(m,n)}=2\mathrm{\parallel }(X_{m-2}+O_{m-2}{)}^{n-2}\mathrm{\parallel }$$ where $2^{m-2}\times 2^{m-2}$ matrices $X_{m-2}$ and $O_{m-2}$ are defined by $$X_{k+1}=\left[\begin{array}{cc}{X}_k& O_k\\ O_k& X_k\end{array}\right]and{O}_{k+1}=\left[\begin{array}{cc}{O}_k& X_k\\ X_k& 4O_k\end{array}\right]$$ for $k=0,1,\cdots ,m-3$, with $1\times 1$ matrices $X_0=\left[\begin{array}{c}1\end{array}\right]$ and $O_0 = \begin{bmatrix} 1 \end{bmatrix}$\mathrm{.}Here$|N|$denotesthesumofallentriesofamatrix$N$. For $n=2$, $(X_{m-2}+O_{m-2}{)}^0$ means the identity matrix of size $2^{m-2} \times 2^{m-2}$.