Minimum lattice length and ropelength of knots

Kyungpyo Hong, Hyoungjun Kim, Seung Sang Oh, Sungjong No

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Let Len(K) be the minimum length of a knot on the cubic lattice (namely the minimum length necessary to construct the knot in the cubic lattice). This paper provides upper bounds for Len(K) of a nontrivial knot K in terms of its crossing number c(K) as follows: Len(K) ≤ min {3/4c(K)<sup>2</sup> + 5c(K) + 17/4, 5/8c(K)<sup>2</sup> + 15/2c(K) + 71/8}. The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. We also provide upper bounds for the minimum ropelength Rop(K) which is close to twice Len(K): Rop(K) ≤ min {1.5c(K)<sup>2</sup> + 9.15c(K) + 6.79, 1.25c(K)<sup>2</sup> + 14.58c(K) + 16.90}.

Original languageEnglish
Article number1460009
JournalJournal of Knot Theory and its Ramifications
Volume23
Issue number7
DOIs
Publication statusPublished - 2014 Jun 25

Keywords

  • Knot
  • lattice knot
  • ropelength
  • upper bound

ASJC Scopus subject areas

  • Algebra and Number Theory

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