Upper bound on the total number of knot n-mosaics

Kyungpyo Hong; Seungsang Oh; Ho Lee; Hwa Jeong Lee

doi:10.1142/S0218216514500655

Upper bound on the total number of knot n-mosaics

Kyungpyo Hong, Seungsang Oh, Ho Lee, Hwa Jeong Lee

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining properly that is called suitably connected. D_n denotes the total number of all knot n-mosaics. Already known is that D₁ = 1, D₂ = 2 and D₃ = 22. In this paper we establish the lower and upper bounds on D_n 2/275 (9 · 6^n-2 + 1)² · 2^(n-3)2 ≤ Dn ≤ 2/275 (9 · 6^n-2 + 1)² · (4.4)^(n-3)2 . and find the exact number of D₄ = 2594.

Original language	English
Article number	1450065
Journal	Journal of Knot Theory and its Ramifications
Volume	23
Issue number	13
DOIs	https://doi.org/10.1142/S0218216514500655
Publication status	Published - 2014 Nov 22

Keywords

Quantum knot
knot mosaic
upper bound

ASJC Scopus subject areas

Algebra and Number Theory

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10.1142/S0218216514500655

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title = "Upper bound on the total number of knot n-mosaics",

abstract = "Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining properly that is called suitably connected. Dn denotes the total number of all knot n-mosaics. Already known is that D1 = 1, D2 = 2 and D3 = 22. In this paper we establish the lower and upper bounds on Dn 2/275 (9 · 6n-2 + 1)2 · 2(n-3)2 ≤ Dn ≤ 2/275 (9 · 6n-2 + 1)2 · (4.4)(n-3)2 . and find the exact number of D4 = 2594.",

keywords = "Quantum knot, knot mosaic, upper bound",

author = "Kyungpyo Hong and Seungsang Oh and Ho Lee and Lee, {Hwa Jeong}",

note = "Funding Information: The second author{\textquoteright}s research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (MSIP) (No. 2011-0021795). Funding Information: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0027989). Publisher Copyright: {\textcopyright} 2014 World Scientific Publishing Company.",

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AU - Hong, Kyungpyo

AU - Oh, Seungsang

AU - Lee, Ho

AU - Lee, Hwa Jeong

N1 - Funding Information: The second author’s research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (MSIP) (No. 2011-0021795). Funding Information: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0027989). Publisher Copyright: © 2014 World Scientific Publishing Company.

PY - 2014/11/22

Y1 - 2014/11/22

N2 - Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining properly that is called suitably connected. Dn denotes the total number of all knot n-mosaics. Already known is that D1 = 1, D2 = 2 and D3 = 22. In this paper we establish the lower and upper bounds on Dn 2/275 (9 · 6n-2 + 1)2 · 2(n-3)2 ≤ Dn ≤ 2/275 (9 · 6n-2 + 1)2 · (4.4)(n-3)2 . and find the exact number of D4 = 2594.

AB - Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining properly that is called suitably connected. Dn denotes the total number of all knot n-mosaics. Already known is that D1 = 1, D2 = 2 and D3 = 22. In this paper we establish the lower and upper bounds on Dn 2/275 (9 · 6n-2 + 1)2 · 2(n-3)2 ≤ Dn ≤ 2/275 (9 · 6n-2 + 1)2 · (4.4)(n-3)2 . and find the exact number of D4 = 2594.

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Upper bound on the total number of knot n-mosaics

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Keywords

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