Ribbonlength of folded ribbon unknots in the plane

TL;DR

This paper investigates the ribbonlength of folded ribbon unknots in the plane, providing bounds and exact values for specific configurations, and discusses related concepts like projection stick number.

Contribution

It introduces bounds and exact minimal ribbonlengths for folded ribbon unknots, particularly for 3-stick unknots, and explores the relationship with projection stick number.

Findings

Upper bound of n cot(π/n) for n-stick unknots

Minimum ribbonlength of 3-stick unknot is 3√3, achieved by an equilateral triangle

Discussion on projection stick number and its relation to ribbonlength

Abstract

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a ribbon, and it turns out that the way the ribbon is folded influences the ribbonlength. We give an upper bound of $n\cot(\pi/n)$ for the ribbonlength of $n$-stick unknots. We prove that the minimum ribbonlength for a 3-stick unknot with the same type of fold at each vertex is $3\sqrt{3}$, and such a minimizer is an equilateral triangle. We end the paper with a discussion of projection stick number and ribbonlength.