Upper bounds on the minimum length of cubic lattice knots

Kyungpyo Hong, Sungjong No, Seungsang Oh

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Knots have been considered to be useful models for simulating molecular chains such as DNA and proteins. One quantity that we are interested in is the molecular knot and the minimum number of monomers necessary to realize a knot. In this paper, we consider every knot in the cubic lattice. In particular, the minimum length of a knot indicates the minimum length necessary to construct the knot in the cubic lattice. Diao introduced this term (he used 'minimum edge number' instead) and proved that the minimum length of the trefoil knot 3 1 is 24. Also the minimum lengths of the knots 41 and 51 are known to be 30 and 34, respectively. In this paper we find a general upper bound of the minimum length of a nontrivial knot K, except the trefoil knot, in terms of the minimum crossing number c(K). The upper bound is 3/2c(K)2-2c(K)+1/2. Moreover, if K is a non-alternating prime knot, then the upper bound is 3/2c(K)2-4c(K)+5/2. Our work can be considered a direct consequence of the results obtained by the authors in Hong et al (2012 arXiv:1209.0048). Furthermore, if K is (n + 1, n)-torus knot, then the upper bound is 6c(k)+2̄c(k)+1+6.

Original languageEnglish
Article number125001
JournalJournal of Physics A: Mathematical and Theoretical
Volume46
Issue number12
DOIs
Publication statusPublished - 2013 Mar 29

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modelling and Simulation
  • Mathematical Physics
  • Physics and Astronomy(all)

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