Period and toroidal knot mosaics

Seungsang Oh, Kyungpyo Hong, Ho Lee, Hwa Jeong Lee, Mi Jeong Yeon

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

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.

Original languageEnglish
Article number1750031
JournalJournal of Knot Theory and its Ramifications
Volume26
Issue number5
DOIs
Publication statusPublished - 2017 Apr 1

Keywords

  • Quantum knot
  • knot mosaic
  • toroidal mosaic

ASJC Scopus subject areas

  • Algebra and Number Theory

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