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Period and toroidal knot mosaics
- Period and toroidal knot mosaics
- Oh, Seungsang; Hong, Kyungpyo; Lee, Ho; Lee, Hwa Jeong; Yeon, Mi Jeong
- DGIST Authors
- Lee, Hwa Jeong
- Issue Date
- Journal of Knot Theory and its Ramifications, 26(5)
- Article Type
- Knot Mosaic; Polynomials; Quantum Knot; Quantum Knots; Toroidal Mosaic
- 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. © 2017 World Scientific Publishing Company.
- World Scientific Publishing Co. Pte Ltd
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