Immersed flat ribbon knots

TL;DR

This paper investigates the minimal ribbonlength of immersed planar ribbon knots and links by embedding their space into disk diagrams, providing examples, bounds, and conjectures related to minimal configurations.

Contribution

It introduces a new approach using disk diagram embeddings to study ribbonlength minimization and explores conditions under which minimizers are ribbon.

Findings

Minimal ribbonlength computed for small knots and links

Examples where minimizers are not ribbon are provided

A bound on crossings for minimal ribbonlength diagrams

Abstract

We study the minimum ribbonlength for immersed planar ribbon knots and links. Our approach is to embed the space of such knots and links into a larger more tractable space of disk diagrams. When length minimisers in disk diagram space are ribbon, then these solve the ribbonlength problem. We also provide examples when minimisers in the space of disk diagrams are not ribbon and state some conjectures. We compute the minimal ribbonlength of some small knot and link diagrams and certain infinite families of link diagrams. Finally we present a bound for the number of crossings for a diagram yielding the minimum ribbonlength of a knot or link amongst all diagrams.