Mosaic number of knots

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

Research output: Contribution to journalArticle

7 Citations (Scopus)

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 6<sup>3</sup><inf>3</inf> link, then m(K) ≤ c(K) - 1.

Original languageEnglish
Article number1450069
JournalJournal of Knot Theory and its Ramifications
Volume23
Issue number13
DOIs
Publication statusPublished - 2014 Nov 22

Keywords

  • knot mosaic
  • mosaic number
  • Quantum knot

ASJC Scopus subject areas

  • Algebra and Number Theory

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