Equilateral stick number of knots

Hyoungjun Kim, Sungjong No, Seung Sang Oh

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

An equilateral stick number s<inf>=</inf>(K) of a knot K is defined to be the minimal number of sticks required to construct a polygonal knot of K which consists of equal length sticks. Rawdon and Scharein [Upper bounds for equilateral stick numbers, in Physical Knots: Knotting, Linking, and Folding Geometric Objects in R<sup>3</sup>, Contemporary Mathematics, Vol. 304 (American Mathematical Society, Providence, RI, 2002), pp. 55-76] found upper bounds for the equilateral stick numbers of all prime knots through 10 crossings by using algorithms in the software KnotPlot. In this paper, we find an upper bound on the equilateral stick number of a non-trivial knot K in terms of the minimal crossing number c(K) which is s<inf>=</inf>(K) ≤ 2c(K) + 2. Moreover if K is a non-alternating prime knot, then s<inf>=</inf>(K) ≤ 2c(K) - 2. Furthermore we find another upper bound on the equilateral stick number for composite knots which is s<inf>=</inf>(K<inf>1</inf>#K<inf>2</inf>) ≤ 2c(K<inf>1</inf>) + 2c(K<inf>2</inf>).

Original language English 1460008 Journal of Knot Theory and its Ramifications 23 7 https://doi.org/10.1142/S0218216514600086 Published - 2014 Jun 25

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Equilateral
Knot
Prime knot
Upper bound
Non-alternating knot
Crossing number
Geometric object
Folding
Composite
Software

Keywords

• equilateral stick number
• Knot
• stick number
• upper bound

ASJC Scopus subject areas

• Algebra and Number Theory

Cite this

Equilateral stick number of knots. / Kim, Hyoungjun; No, Sungjong; Oh, Seung Sang.

In: Journal of Knot Theory and its Ramifications, Vol. 23, No. 7, 1460008, 25.06.2014.

Research output: Contribution to journalArticle

Kim, Hyoungjun ; No, Sungjong ; Oh, Seung Sang. / Equilateral stick number of knots. In: Journal of Knot Theory and its Ramifications. 2014 ; Vol. 23, No. 7.
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