Ribbonlength bounds for pretzel links and knots with $\leq 9$ crossings

Abstract

Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded ribbon knots, which is defined as the knot's length-to-width ratio. The {\em ribbonlength problem} asks to find the infimal folded ribbonlength of a knot or link type. We prove that any $P(p,q,r)$ pretzel link can be constructed so that its infimal folded ribbonlength is $\leq \frac{55}{\sqrt{3}} \leq 31.755$. We prove that any $n$-strand pretzel link $P(p_1,p_2, \dots, p_n)$ can be constructed so that its infimal folded ribbonlength is $\leq \frac{18n+1}{\sqrt{3}}$. This means that there is an infinite link family with a uniform bound on infimal folded ribbonlength. That is, we have shown $\alpha=0$ in the equation $c\cdot \text{Cr}(L)^\alpha \leq \text{Rib}([L])$, where $L$ is any link and $c$ is a constant. This paper also contains a table showing the best known upper bounds on the infimal folded ribbonlength for all knots with $\leq 9$ crossings.