The discovery of the Jones polynomial made an important connection between quantum physics and knot theory. Kauffman and Lomonaco introduced the knot mosaic system to define the quantum knot system for the purpose of representing an actual physical quantum system. This paper is inspired by an open question about the knot mosaic enumeration suggested by them. A knot [Formula: see text]-mosaic is an [Formula: see text] array of 11 mosaic tiles representing a knot diagram by adjoining properly. The total number [Formula: see text] of knot [Formula: see text]-mosaics, which indicates the dimension of the Hilbert space of the quantum knot system, is known to grow in a quadratic exponential rate. Recently, Oh et al. developed the state matrix recursion method producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. Furthermore, they showed the existence of the knot mosaic constant [Formula: see text] and found its upper and lower bounds in a series of papers. The latest upper bound was obtained through two new concepts: quasimosaics and cling mosaics. As a sequel to this research program, we adjust the state matrix recursion method to handle cling mosaics inside a quasimosaic, which is called the progressive state matrix recursion method. This method provides recursive matrix-relations producing a sharper bound of the knot mosaic constant: [Formula: see text]The discovery of the Jones polynomial made an important connection between quantum physics and knot theory. Kauffman and Lomonaco introduced the knot mosaic system to define the quantum knot system for the purpose of representing an actual physical quantum system. This paper is inspired by an open question about the knot mosaic enumeration suggested by them. A knot [Formula: see text]-mosaic is an [Formula: see text] array of 11 mosaic tiles representing a knot diagram by adjoining properly. The total number [Formula: see text] of knot [Formula: see text]-mosaics, which indicates the dimension of the Hilbert space of the quantum knot system, is known to grow in a quadratic exponential rate. Recently, Oh et al. developed the state matrix recursion method producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. Furthermore, they showed the existence of the knot mosaic constant [Formula: see text] and found its upper and lower bounds in a series of papers. The latest upper bound was obtained through two new concepts: quasimosaics and cling mosaics. As a sequel to this research program, we adjust the state matrix recursion method to handle cling mosaics inside a quasimosaic, which is called the progressive state matrix recursion method. This method provides recursive matrix-relations producing a sharper bound of the knot mosaic constant: [Formula: see text]

Reviews in Mathematical PhysicsApr 11, 20240 citations

Quantum Knots Revisited

Samuel J. Lomonaco

Jan 1, 20180 citations

Quantum knots and the number of knot mosaics

Seungsang Oh, Kyungpyo Hong, Ho Lee +1 more

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 $2^{m-2} \times 2^{m-2}$ .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 $2^{m-2} \times 2^{m-2}$.

arXiv:1412.4460Dec 15, 20141 citation

Quantum knots revisited (Conference Presentation)

Samuel J. Lomonaco

SPIE Quantum Information Science, Sensing, and ComputationMay 14, 20180 citations

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}