An upper bound on stick numbers of knots

TL;DR

This paper improves the known upper bound on the stick number of knots, relating it more tightly to the minimal crossing number, especially for non-alternating prime knots.

Contribution

It presents a tighter upper bound on the stick number of knots, refining Negami's 1991 bound, with specific improvements for non-alternating prime knots.

Findings

Improved upper bound: s(K) ≤ 1.5(c(K)+1)

For non-alternating prime knots: s(K) ≤ 1.5 c(K)

Enhances understanding of knot complexity measures

Abstract

In 1991, Negami found an upper bound on the stick number $s(K)$ of a nontrivial knot $K$ in terms of the minimal crossing number $c(K)$ of the knot which is $s(K) \leq 2 c(K)$. In this paper we improve this upper bound to $s(K) \leq \frac{3}{2} (c(K)+1)$. Moreover if $K$ is a non-alternating prime knot, then $s(K) \leq \frac{3}{2} c(K)$.