Period and toroidal knot mosaics

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

Research output: Contribution to journalArticle

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 (Formula presented.)-mosaic is an (Formula presented.) 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 (Formula presented.)-mosaics for any positive integers (Formula presented.) and (Formula presented.), toroidal knot (Formula presented.)-mosaics for co-prime integers (Formula presented.) and (Formula presented.), and furthermore toroidal knot (Formula presented.)-mosaics for a prime number (Formula presented.). We also analyze the asymptotics of the growth rates of their cardinality.

Original languageEnglish
JournalJournal of Knot Theory and its Ramifications
DOIs
Publication statusAccepted/In press - 2017

    Fingerprint

Keywords

  • knot mosaic
  • Quantum knot
  • toroidal mosaic

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this