Upper bound on the total number of knot n-mosaics

Kyungpyo Hong, Seung Sang Oh, Ho Lee, Hwa Jeong Lee

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of 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 by adjoining properly that is called suitably connected. D<inf>n</inf> denotes the total number of all knot n-mosaics. Already known is that D<inf>1</inf> = 1, D<inf>2</inf> = 2 and D<inf>3</inf> = 22. In this paper we establish the lower and upper bounds on D<inf>n</inf> 2/275 (9 · 6<sup>n-2</sup> + 1)<sup>2</sup> · 2<sup>(n-3)2</sup> ≤ Dn ≤ 2/275 (9 · 6<sup>n-2</sup> + 1)<sup>2</sup> · (4.4)<sup>(n-3)2</sup> . and find the exact number of D<inf>4</inf> = 2594.

Original languageEnglish
Article number1450065
JournalJournal of Knot Theory and its Ramifications
Volume23
Issue number13
DOIs
Publication statusPublished - 2014 Nov 22

Keywords

  • knot mosaic
  • Quantum knot
  • upper bound

ASJC Scopus subject areas

  • Algebra and Number Theory

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