Equilateral stick number of knots

Abstract

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 [12] 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 nontrivial knot K in terms of the minimal crossing number $c(K)$ which is $s_{=}(K) \le 2c(K) + 2$. Moreover if $K$ is a non-alternating prime knot, then $s_{=}(K) \le 2c(K) - 2$. Furthermore we find another upper bound on the equilateral stick number for composite knots which is $s_{=}(K_1 \# K_2) \le 2c(K_1) + 2c(K_2)$.