TY - JOUR
T1 - Equilateral stick number of knots
AU - Kim, Hyoungjun
AU - No, Sungjong
AU - Oh, Seungsang
N1 - Funding Information:
This 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). This 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 - 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 - Knot
KW - equilateral stick number
KW - stick number
KW - upper bound
UR - http://www.scopus.com/inward/record.url?scp=84928372397&partnerID=8YFLogxK
U2 - 10.1142/S0218216514600086
DO - 10.1142/S0218216514600086
M3 - Article
AN - SCOPUS:84928372397
SN - 0218-2165
VL - 23
JO - Journal of Knot Theory and its Ramifications
JF - Journal of Knot Theory and its Ramifications
IS - 7
M1 - 1460008
ER -