### Abstract

Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m, n)-mosaic is an m x n matrix of mosaic tiles which are T_{0} through T_{10} depicted, representing a knot or a link by adjoining properly that is called suitably connected. An interesting question in studying mosaic theory is how many knot (m, n)-mosaics are there. D_{m,n}denotes the total number of all knot (m, n)-mosaics. This counting is very important because the total number of knot mosaics is indeed the dimension of the Hilbert space of these quantum knot mosaics. In this paper, we find a table of the precise values of D_{m,n} for 4 ≤ m ≤ n ≤ 6. Mainly we use a partition matrix argument which turns out to be remarkably efficient to count small knot mosaics.

Original language | English |
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Article number | 435201 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 47 |

Issue number | 43 |

DOIs | |

Publication status | Published - 2014 Oct 31 |

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### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*47*(43), [435201]. https://doi.org/10.1088/1751-8113/47/43/435201

**Small knot mosaics and partition matrices.** / Hong, Kyungpyo; Lee, Ho; Lee, Hwa Jeong; Oh, Seung Sang.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 47, no. 43, 435201. https://doi.org/10.1088/1751-8113/47/43/435201

}

TY - JOUR

T1 - Small knot mosaics and partition matrices

AU - Hong, Kyungpyo

AU - Lee, Ho

AU - Lee, Hwa Jeong

AU - Oh, Seung Sang

PY - 2014/10/31

Y1 - 2014/10/31

N2 - Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m, n)-mosaic is an m x n matrix of mosaic tiles which are T0 through T10 depicted, representing a knot or a link by adjoining properly that is called suitably connected. An interesting question in studying mosaic theory is how many knot (m, n)-mosaics are there. Dm,ndenotes the total number of all knot (m, n)-mosaics. This counting is very important because the total number of knot mosaics is indeed the dimension of the Hilbert space of these quantum knot mosaics. In this paper, we find a table of the precise values of Dm,n for 4 ≤ m ≤ n ≤ 6. Mainly we use a partition matrix argument which turns out to be remarkably efficient to count small knot mosaics.

AB - Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m, n)-mosaic is an m x n matrix of mosaic tiles which are T0 through T10 depicted, representing a knot or a link by adjoining properly that is called suitably connected. An interesting question in studying mosaic theory is how many knot (m, n)-mosaics are there. Dm,ndenotes the total number of all knot (m, n)-mosaics. This counting is very important because the total number of knot mosaics is indeed the dimension of the Hilbert space of these quantum knot mosaics. In this paper, we find a table of the precise values of Dm,n for 4 ≤ m ≤ n ≤ 6. Mainly we use a partition matrix argument which turns out to be remarkably efficient to count small knot mosaics.

KW - Knot mosaic

KW - Partition matrix

KW - Quantum knot

KW - Quantum physics

UR - http://www.scopus.com/inward/record.url?scp=84908360021&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908360021&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/47/43/435201

DO - 10.1088/1751-8113/47/43/435201

M3 - Article

VL - 47

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 43

M1 - 435201

ER -