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Published in:
01-03-2015
Quantum knots and the number of knot mosaics
Published in: Quantum Information Processing | Issue 3/2015
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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. where \(2^{m-2} \times 2^{m-2}\) matrices \(X_{m-2}\) and \(O_{m-2}\) are defined by 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 \(2^{m-2} \times 2^{m-2}\).
$$\begin{aligned} D^{(m,n)} = 2 \, \Vert (X_{m-2}+O_{m-2})^{n-2} \Vert \end{aligned}$$
$$\begin{aligned} X_{k+1} = \begin{bmatrix} X_k&O_k \\ O_k&X_k \end{bmatrix} \ \hbox {and } \ O_{k+1} = \begin{bmatrix} O_k&X_k \\ X_k&4 \, O_k \end{bmatrix} \end{aligned}$$