Period and toroidal knot mosaics

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

doi:10.1142/S0218216517500316

Period and toroidal knot mosaics

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

^*Corresponding author for this work

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

6 Citations (Scopus)

Abstract

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on 'Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an m × n matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot (m,n)-mosaics for any positive integers m and n, toroidal knot (m,n)-mosaics for co-prime integers m and n, and furthermore toroidal knot (p,p)-mosaics for a prime number p. We also analyze the asymptotics of the growth rates of their cardinality.

Original language	English
Article number	1750031
Journal	Journal of Knot Theory and its Ramifications
Volume	26
Issue number	5
DOIs	https://doi.org/10.1142/S0218216517500316
Publication status	Published - 2017 Apr 1

Bibliographical note

Funding Information:
The corresponding author(Seungsang Oh) was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. NRF-2014R1A2A1A11050999). Hwa Jeong Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF- 2015R1C1A2A01054607).

Publisher Copyright:
© 2017 World Scientific Publishing Company.

Keywords

Quantum knot
knot mosaic
toroidal mosaic

ASJC Scopus subject areas

Algebra and Number Theory

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10.1142/S0218216517500316

Cite this

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title = "Period and toroidal knot mosaics",

abstract = "Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on 'Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an m × n matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot (m,n)-mosaics for any positive integers m and n, toroidal knot (m,n)-mosaics for co-prime integers m and n, and furthermore toroidal knot (p,p)-mosaics for a prime number p. We also analyze the asymptotics of the growth rates of their cardinality.",

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

note = "Funding Information: The corresponding author(Seungsang Oh) was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. NRF-2014R1A2A1A11050999). Hwa Jeong Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT \& Future Planning (NRF- 2015R1C1A2A01054607). Publisher Copyright: {\textcopyright} 2017 World Scientific Publishing Company.",

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doi = "10.1142/S0218216517500316",

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N1 - Funding Information: The corresponding author(Seungsang Oh) was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. NRF-2014R1A2A1A11050999). Hwa Jeong Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF- 2015R1C1A2A01054607). Publisher Copyright: © 2017 World Scientific Publishing Company.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on 'Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an m × n matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot (m,n)-mosaics for any positive integers m and n, toroidal knot (m,n)-mosaics for co-prime integers m and n, and furthermore toroidal knot (p,p)-mosaics for a prime number p. We also analyze the asymptotics of the growth rates of their cardinality.

AB - Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on 'Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an m × n matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot (m,n)-mosaics for any positive integers m and n, toroidal knot (m,n)-mosaics for co-prime integers m and n, and furthermore toroidal knot (p,p)-mosaics for a prime number p. We also analyze the asymptotics of the growth rates of their cardinality.

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KW - toroidal mosaic

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SN - 0218-2165

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JF - Journal of Knot Theory and its Ramifications

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M1 - 1750031

ER -

Period and toroidal knot mosaics

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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