Stick numbers of $2$-bridge knots and links

TL;DR

This paper improves the upper bound on the stick number for 2-bridge knots and links, showing they can be constructed with only c(K)+2 sticks, which is tighter than previous bounds.

Contribution

It introduces a new construction method demonstrating that any 2-bridge knot or link with at least six crossings can be built with c(K)+2 sticks, refining existing bounds.

Findings

Constructed 2-bridge knots/links with c(K)+2 sticks

Established a tighter upper bound on stick numbers

Applicable to knots with at least six crossings

Abstract

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 $s(K) \leq 2 c(K)$. Furthermore McCabe proved $s(K) \leq c(K) + 3$ for a $2$-bridge knot or link, except in the case of the unlink and the Hopf link. In this paper we construct any $2$-bridge knot or link $K$ of at least six crossings by using only $c(K)+2$ straight sticks. This gives a new upper bound on stick numbers of $2$-bridge knots and links in terms of crossing numbers.