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).

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

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U2 - 10.1142/S0218216514600086

DO - 10.1142/S0218216514600086

M3 - Article

AN - SCOPUS:84928372397

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 -