### Abstract

For a certain infinite family F of knots or links, we study the growth power ratios of their stick number, lattice stick number, minimum lattice length and minimum ropelength compared with their minimum crossing number c(K) for every K ∈ F. It is known that the stick number and lattice stick number grow between the 1/2 and linear power of the crossing number, and minimum lattice length and minimum ropelength grow with at least the 3/4 power of crossing number (which is called the four-thirds power law). Furthermore, the minimal lattice length and minimum ropelength grow at most as O (c (K)[ln(c (K))]5), but it is unknown whether any family exhibits superlinear growth. For any real number r between 1/2 and 1, we give an infinite family of non-splittable prime links in which the stick number and lattice stick number grow exactly as the rth power of crossing number. Furthermore for any real number r between 3/4 and 1, we give another infinite family of non-splittable prime links in which the minimum lattice length and minimum ropelength grow exactly as the rth power of crossing number.

Original language | English |
---|---|

Article number | 035202 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 48 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2015 Jan 23 |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*48*(3), [035202]. https://doi.org/10.1088/1751-8113/48/3/035202

**Link lengths and their growth powers.** / Huh, Youngsik; No, Sungjong; Oh, Seung Sang; Rawdon, Eric J.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 48, no. 3, 035202. https://doi.org/10.1088/1751-8113/48/3/035202

}

TY - JOUR

T1 - Link lengths and their growth powers

AU - Huh, Youngsik

AU - No, Sungjong

AU - Oh, Seung Sang

AU - Rawdon, Eric J.

PY - 2015/1/23

Y1 - 2015/1/23

N2 - For a certain infinite family F of knots or links, we study the growth power ratios of their stick number, lattice stick number, minimum lattice length and minimum ropelength compared with their minimum crossing number c(K) for every K ∈ F. It is known that the stick number and lattice stick number grow between the 1/2 and linear power of the crossing number, and minimum lattice length and minimum ropelength grow with at least the 3/4 power of crossing number (which is called the four-thirds power law). Furthermore, the minimal lattice length and minimum ropelength grow at most as O (c (K)[ln(c (K))]5), but it is unknown whether any family exhibits superlinear growth. For any real number r between 1/2 and 1, we give an infinite family of non-splittable prime links in which the stick number and lattice stick number grow exactly as the rth power of crossing number. Furthermore for any real number r between 3/4 and 1, we give another infinite family of non-splittable prime links in which the minimum lattice length and minimum ropelength grow exactly as the rth power of crossing number.

AB - For a certain infinite family F of knots or links, we study the growth power ratios of their stick number, lattice stick number, minimum lattice length and minimum ropelength compared with their minimum crossing number c(K) for every K ∈ F. It is known that the stick number and lattice stick number grow between the 1/2 and linear power of the crossing number, and minimum lattice length and minimum ropelength grow with at least the 3/4 power of crossing number (which is called the four-thirds power law). Furthermore, the minimal lattice length and minimum ropelength grow at most as O (c (K)[ln(c (K))]5), but it is unknown whether any family exhibits superlinear growth. For any real number r between 1/2 and 1, we give an infinite family of non-splittable prime links in which the stick number and lattice stick number grow exactly as the rth power of crossing number. Furthermore for any real number r between 3/4 and 1, we give another infinite family of non-splittable prime links in which the minimum lattice length and minimum ropelength grow exactly as the rth power of crossing number.

KW - minimum lattice length

KW - ropelength

KW - stick number

UR - http://www.scopus.com/inward/record.url?scp=84920064585&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84920064585&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/48/3/035202

DO - 10.1088/1751-8113/48/3/035202

M3 - Article

AN - SCOPUS:84920064585

VL - 48

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 3

M1 - 035202

ER -