Mosaic number of knots

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

doi:10.1142/S0218216514500692

Mosaic number of knots

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

^*Corresponding author for this work

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

16 Citations (Scopus)

Abstract

Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent 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. The mosaic number m(K) of a knot K is the smallest integer n for which K is representable as a knot n-mosaic. In this paper, we establish an upper bound on the mosaic number of a knot or a link K in terms of the crossing number c(K). Let K be a nontrivial knot or a non-split link except the Hopf link. Then m(K) ≤ c(K) + 1. Moreover if K is prime and non-alternating except 6³₃ link, then m(K) ≤ c(K) - 1.

Original language	English
Article number	1450069
Journal	Journal of Knot Theory and its Ramifications
Volume	23
Issue number	13
DOIs	https://doi.org/10.1142/S0218216514500692
Publication status	Published - 2014 Nov 22

Bibliographical note

Funding Information:
The corresponding author (Seungsang Oh) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (MSIP) (No. 2011-0021795). 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.

Keywords

Quantum knot
knot mosaic
mosaic number

ASJC Scopus subject areas

Algebra and Number Theory

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10.1142/S0218216514500692

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title = "Mosaic number of knots",

abstract = "Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent 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. The mosaic number m(K) of a knot K is the smallest integer n for which K is representable as a knot n-mosaic. In this paper, we establish an upper bound on the mosaic number of a knot or a link K in terms of the crossing number c(K). Let K be a nontrivial knot or a non-split link except the Hopf link. Then m(K) ≤ c(K) + 1. Moreover if K is prime and non-alternating except 633 link, then m(K) ≤ c(K) - 1.",

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note = "Funding Information: The corresponding author (Seungsang Oh) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT \& Future Planning (MSIP) (No. 2011-0021795). 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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N2 - Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent 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. The mosaic number m(K) of a knot K is the smallest integer n for which K is representable as a knot n-mosaic. In this paper, we establish an upper bound on the mosaic number of a knot or a link K in terms of the crossing number c(K). Let K be a nontrivial knot or a non-split link except the Hopf link. Then m(K) ≤ c(K) + 1. Moreover if K is prime and non-alternating except 633 link, then m(K) ≤ c(K) - 1.

AB - Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent 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. The mosaic number m(K) of a knot K is the smallest integer n for which K is representable as a knot n-mosaic. In this paper, we establish an upper bound on the mosaic number of a knot or a link K in terms of the crossing number c(K). Let K be a nontrivial knot or a non-split link except the Hopf link. Then m(K) ≤ c(K) + 1. Moreover if K is prime and non-alternating except 633 link, then m(K) ≤ c(K) - 1.

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KW - mosaic number

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Mosaic number of knots

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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