Minimum lattice length and ropelength of knots

Kyungpyo Hong; Hyoungjun Kim; Seungsang Oh; Sungjong No

doi:10.1142/S0218216514600098

Minimum lattice length and ropelength of knots

Kyungpyo Hong, Hyoungjun Kim, Seungsang Oh, Sungjong No

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

6 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)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}.

Original language	English
Article number	1460009
Journal	Journal of Knot Theory and its Ramifications
Volume	23
Issue number	7
DOIs	https://doi.org/10.1142/S0218216514600098
Publication status	Published - 2014 Jun 25

Keywords

Knot
lattice knot
ropelength
upper bound

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.1142/S0218216514600098

Cite this

@article{48d41ed051164132b74f07eeea3bce17,

title = "Minimum lattice length and ropelength of knots",

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)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}.",

keywords = "Knot, lattice knot, ropelength, upper bound",

author = "Kyungpyo Hong and Hyoungjun Kim and Seungsang Oh and Sungjong No",

note = "Funding Information: The first three authors{\textquoteright} research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (MSIP) (No. 2011-0021795). The fourth author{\textquoteright}s work was supported by the BK21 Plus Project through the National Research Foundation of Korea (NRF) funded by the Korean Ministry of Education (22A20130011003). Publisher Copyright: {\textcopyright} 2014 World Scientific Publishing Company.",

year = "2014",

month = jun,

day = "25",

doi = "10.1142/S0218216514600098",

language = "English",

volume = "23",

journal = "Journal of Knot Theory and its Ramifications",

issn = "0218-2165",

publisher = "World Scientific Publishing Co. Pte Ltd",

number = "7",

}

TY - JOUR

T1 - Minimum lattice length and ropelength of knots

AU - Hong, Kyungpyo

AU - Kim, Hyoungjun

AU - Oh, Seungsang

AU - No, Sungjong

N1 - Funding Information: The first three authors’ research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (MSIP) (No. 2011-0021795). The fourth author’s work was supported by the BK21 Plus Project through the National Research Foundation of Korea (NRF) funded by the Korean Ministry of Education (22A20130011003). Publisher Copyright: © 2014 World Scientific Publishing Company.

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

U2 - 10.1142/S0218216514600098

DO - 10.1142/S0218216514600098

M3 - Article

AN - SCOPUS:84928361266

SN - 0218-2165

VL - 23

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

IS - 7

M1 - 1460009

ER -

Minimum lattice length and ropelength of knots

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this