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- W2951184582 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 $m,n leq 6$ by the authors. In this paper, we construct an algorithm producing the precise value of $D^{(m,n)}$ for $m,n geq 2$ that uses recurrence relations of state matrices that turn out to be remarkably efficient to count knot mosaics. $$ D^{(m,n)} = 2 , | (X_{m-2}+O_{m-2})^{n-2} | $$ where $2^{m-2} times 2^{m-2}$ matrices $X_{m-2}$ and $O_{m-2}$ are defined by $$ X_{k+1} = begin{bmatrix} X_k & O_k O_k & X_k end{bmatrix} mbox{and } O_{k+1} = begin{bmatrix} O_k & X_k X_k & 4 , O_k end{bmatrix} $$ 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 $|N|$ 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 $2^{m-2} times 2^{m-2}$." @default.
- W2951184582 created "2019-06-27" @default.
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- W2951184582 date "2014-12-15" @default.
- W2951184582 modified "2023-09-23" @default.
- W2951184582 title "Quantum knots and the number of knot mosaics" @default.
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