### 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 |
---|---|

Article number | 1460008 |

Journal | Journal of Knot Theory and its Ramifications |

Volume | 23 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2014 Jun 25 |

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### Keywords

- equilateral stick number
- Knot
- stick number
- upper bound

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Knot Theory and its Ramifications*,

*23*(7), [1460008]. https://doi.org/10.1142/S0218216514600086

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

Research output: Contribution to journal › Article

*Journal of Knot Theory and its Ramifications*, vol. 23, no. 7, 1460008. https://doi.org/10.1142/S0218216514600086

}

TY - JOUR

T1 - Equilateral stick number of knots

AU - Kim, Hyoungjun

AU - No, Sungjong

AU - Oh, Seung Sang

PY - 2014/6/25

Y1 - 2014/6/25

N2 - An equilateral stick number s=(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 R3, 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=(K) ≤ 2c(K) + 2. Moreover if K is a non-alternating prime knot, then s=(K) ≤ 2c(K) - 2. Furthermore we find another upper bound on the equilateral stick number for composite knots which is s=(K1#K2) ≤ 2c(K1) + 2c(K2).

AB - An equilateral stick number s=(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 R3, 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=(K) ≤ 2c(K) + 2. Moreover if K is a non-alternating prime knot, then s=(K) ≤ 2c(K) - 2. Furthermore we find another upper bound on the equilateral stick number for composite knots which is s=(K1#K2) ≤ 2c(K1) + 2c(K2).

KW - equilateral stick number

KW - Knot

KW - stick number

KW - upper bound

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U2 - 10.1142/S0218216514600086

DO - 10.1142/S0218216514600086

M3 - Article

VL - 23

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 7

M1 - 1460008

ER -