### 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 language | English |
---|---|

Article number | 1460009 |

Journal | Journal of Knot Theory and its Ramifications |

Volume | 23 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2014 Jun 25 |

### Fingerprint

### Keywords

- Knot
- lattice knot
- ropelength
- upper bound

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Knot Theory and its Ramifications*,

*23*(7), [1460009]. https://doi.org/10.1142/S0218216514600098

**Minimum lattice length and ropelength of knots.** / Hong, Kyungpyo; Kim, Hyoungjun; Oh, Seung Sang; No, Sungjong.

Research output: Contribution to journal › Article

*Journal of Knot Theory and its Ramifications*, vol. 23, no. 7, 1460009. https://doi.org/10.1142/S0218216514600098

}

TY - JOUR

T1 - Minimum lattice length and ropelength of knots

AU - Hong, Kyungpyo

AU - Kim, Hyoungjun

AU - Oh, Seung Sang

AU - No, Sungjong

PY - 2014/6/25

Y1 - 2014/6/25

N2 - 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)2 + 5c(K) + 17/4, 5/8c(K)2 + 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)2 + 9.15c(K) + 6.79, 1.25c(K)2 + 14.58c(K) + 16.90}.

AB - 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)2 + 5c(K) + 17/4, 5/8c(K)2 + 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)2 + 9.15c(K) + 6.79, 1.25c(K)2 + 14.58c(K) + 16.90}.

KW - Knot

KW - lattice knot

KW - ropelength

KW - upper bound

UR - http://www.scopus.com/inward/record.url?scp=84928361266&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928361266&partnerID=8YFLogxK

U2 - 10.1142/S0218216514600098

DO - 10.1142/S0218216514600098

M3 - Article

AN - SCOPUS:84928361266

VL - 23

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 7

M1 - 1460009

ER -