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- Title
Quantum knots and the number of knot mosaics.
- Authors
Oh, Seungsang; Hong, Kyungpyo; Lee, Ho; Lee, Hwa
- 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 $$2^{m-2} \times 2^{m-2}$$ .
- Subjects
QUANTUM numbers; KNOT theory; HILBERT space; QUANTUM theory; DIMENSIONS
- Publication
Quantum Information Processing, 2015, Vol 14, Issue 3, p801
- ISSN
1570-0755
- Publication type
Article
- DOI
10.1007/s11128-014-0895-7