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 | |
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