Linking number and folded ribbon unknots

TL;DR

This paper investigates the minimal folded ribbonlength of folded ribbon knots, particularly unknots with various linking numbers, establishing bounds and optimal configurations for polygons with obtuse angles.

Contribution

It determines the minimum folded ribbonlength for 3-stick unknots with specific linking numbers and proves that regular polygons minimize ribbonlength among n-gons with obtuse angles.

Findings

Minimum folded ribbonlength for 3-stick unknots with linking numbers ±1 and ±3.

Optimal n-gon configuration for minimal ribbonlength is regular.

Upper bound of 2n for folded ribbon unknot with ribbon linking number ±n.

Abstract

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a folded ribbon knot. The folded ribbon knot is also a framed knot, and the ribbon linking number is the linking number of the knot and one boundary component of the ribbon. We find the minimum folded ribbonlength for $3$-stick unknots with ribbon linking numbers $\pm1$ and $\pm 3$, and we prove that the minimum folded ribbonlength for $n$-gons with obtuse interior angles is achieved when the $n$-gon is regular. Among other results, we prove that the minimum folded ribbonlength of any folded ribbon unknot which is a topological annulus with ribbon linking number $\pm n$ is bounded from above by $2n$.