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Growth rate of quantum knot mosaics

Authors
Oh, SeungsangKim, Youngin
Issue Date
8월-2019
Publisher
SPRINGER
Keywords
Quantum knot; Knot mosaic; Mosaic growth rate
Citation
QUANTUM INFORMATION PROCESSING, v.18, no.8
Indexed
SCIE
SCOPUS
Journal Title
QUANTUM INFORMATION PROCESSING
Volume
18
Number
8
URI
https://scholar.korea.ac.kr/handle/2021.sw.korea/63615
DOI
10.1007/s11128-019-2353-z
ISSN
1570-0755
Abstract
Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. This paper is inspired by an open question about the knot mosaic enumeration suggested by them. A knot (m,n)-mosaic is an mxn array of 11 mosaic tiles representing a knot or a link diagram by adjoining properly. The total number Dm,n of knot (m,n)-mosaics, which indicates the dimension of the Hilbert space of the quantum knot system, is known to grow in a quadratic exponential rate. Recently, the first author showed the existence of the knot mosaic constant and proved 44.302 by developing an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. In this paper, we give a simpler proof of the lower bound and improve the upper bound of the knot mosaic constant as 4 <= d = 4.113 ... by introducing two new concepts: quasimosaics and cling mosaics.
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