Folded ribbon knots in the plane

TL;DR

This survey explores folded ribbon knots in the plane, focusing on ribbonlength minimization, known bounds, and how folding choices influence the minimal length for various knot types.

Contribution

It summarizes existing results on ribbonlength bounds, discusses the impact of folding choices, and compares different folded ribbon equivalence types.

Findings

Upper bounds for twist and torus knots' ribbonlength

Ribbonlength bounds relate to crossing number

Fold choice affects ribbonlength

Abstract

This survey reviews 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 the ribbonlength problem asks to minimize the ribbonlength for a given knot type. We give a summary of known results. For the most part, these are upper bounds of ribbonlength of twist knots and certain families of torus knots. We discuss result of G. Tian, which give upper bounds of ribbonlength in terms of crossing number. In addition, it turns out the choice of fold affects the ribbonlength. We end with a discussion of three different types of folded ribbon equivalence and give examples illustrating their relationship to ribbonlength.