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

TL;DR

This paper establishes upper bounds on the folded ribbonlength for pretzel links and knots with up to nine crossings, demonstrating that certain infinite families have bounded ribbonlength regardless of crossing number.

Contribution

It provides explicit upper bounds for the infimal folded ribbonlength of pretzel links and knots with up to nine crossings, including infinite families with uniform bounds.

Findings

Bounded ribbonlength for all pretzel links with specific parameters.

Infinite link families with uniform upper bounds on ribbonlength.

Table of best known bounds for knots with ≤ 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.