Quantum knots and the number of knot mosaics

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 $$(m,n)$$ ( m , n ) -mosaic is an $$m \times n$$ m × n matrix of mosaic tiles ( $$T_0$$ T 0 through $$T_{10}$$ T 10 depicted in the introduction) representing a knot or a link by adjoining properly that is called suitably connected. $$D^{(m,n)}$$ D ( m , n ) is the total number of all knot $$(m,n)$$ ( m , n ) -mosaics. This value indicates the dimension of the Hilbert space of these quantum knot system. $$D^{(m,n)}$$ D ( m , n ) is already found for $$m,n \le 6$$ m , n ≤ 6 by the authors. In this paper, we construct an algorithm producing the precise value of $$D^{(m,n)}$$ D ( m , n ) for $$m,n \ge 2$$ m , n ≥ 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} D ( m , n ) = 2 ‖ ( X m - 2 + O m - 2 ) n - 2 ‖ where $$2^{m-2} \times 2^{m-2}$$ 2 m - 2 × 2 m - 2 matrices $$X_{m-2}$$ X m - 2 and $$O_{m-2}$$ 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} X k + 1 = X k O k O k X k and O k + 1 = O k X k X k 4 O k for $$k=0,1, \cdots , m-3$$ k = 0 , 1 , ⋯ , m - 3 , with $$1 \times 1$$ 1 × 1 matrices $$X_0 = \begin{bmatrix} 1 \end{bmatrix}$$ X 0 = 1 and $$O_0 = \begin{bmatrix} 1 \end{bmatrix}$$ O 0 = 1 . Here $$\Vert N\Vert$$ ‖ N ‖ denotes the sum of all entries of a matrix $$N$$ N . For $$n=2$$ n = 2 , $$(X_{m-2}+O_{m-2})^0$$ ( X m - 2 + O m - 2 ) 0 means the identity matrix of size $$2^{m-2} \times 2^{m-2}$$ 2 m - 2 × 2 m - 2 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

Quantum knots and the number of knot mosaics

, Volume 14 (3) – Dec 12, 2014
11 pages

/lp/springer_journal/quantum-knots-and-the-number-of-knot-mosaics-u0LHqYdqgn
Publisher
Springer US
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-014-0895-7
Publisher site
See Article on Publisher Site

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 $$(m,n)$$ ( m , n ) -mosaic is an $$m \times n$$ m × n matrix of mosaic tiles ( $$T_0$$ T 0 through $$T_{10}$$ T 10 depicted in the introduction) representing a knot or a link by adjoining properly that is called suitably connected. $$D^{(m,n)}$$ D ( m , n ) is the total number of all knot $$(m,n)$$ ( m , n ) -mosaics. This value indicates the dimension of the Hilbert space of these quantum knot system. $$D^{(m,n)}$$ D ( m , n ) is already found for $$m,n \le 6$$ m , n ≤ 6 by the authors. In this paper, we construct an algorithm producing the precise value of $$D^{(m,n)}$$ D ( m , n ) for $$m,n \ge 2$$ m , n ≥ 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} D ( m , n ) = 2 ‖ ( X m - 2 + O m - 2 ) n - 2 ‖ where $$2^{m-2} \times 2^{m-2}$$ 2 m - 2 × 2 m - 2 matrices $$X_{m-2}$$ X m - 2 and $$O_{m-2}$$ 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} X k + 1 = X k O k O k X k and O k + 1 = O k X k X k 4 O k for $$k=0,1, \cdots , m-3$$ k = 0 , 1 , ⋯ , m - 3 , with $$1 \times 1$$ 1 × 1 matrices $$X_0 = \begin{bmatrix} 1 \end{bmatrix}$$ X 0 = 1 and $$O_0 = \begin{bmatrix} 1 \end{bmatrix}$$ O 0 = 1 . Here $$\Vert N\Vert$$ ‖ N ‖ denotes the sum of all entries of a matrix $$N$$ N . For $$n=2$$ n = 2 , $$(X_{m-2}+O_{m-2})^0$$ ( X m - 2 + O m - 2 ) 0 means the identity matrix of size $$2^{m-2} \times 2^{m-2}$$ 2 m - 2 × 2 m - 2 .

Journal

Quantum Information ProcessingSpringer Journals

Published: Dec 12, 2014

References

• On upper bounds for toroidal mosaic numbers
Carlisle, MJ; Laufer, MS

DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month Explore the DeepDyve Library Unlimited reading Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere. Stay up to date Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates. Organize your research It’s easy to organize your research with our built-in tools. Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve Freelancer DeepDyve Pro Price FREE$49/month

\$360/year
Save searches from