Profiles of Critical Flat Ribbon Knots

TL;DR

This paper computes minimal ribbonlengths for various knots, including complex and infinite classes, contributing to the understanding of geometric knot optimization and proposing conjectures linking ribbonlength to knot invariants.

Contribution

It provides the first calculations of minimal ribbonlength for several well-known and infinite classes of knots, advancing the metric classification of knots.

Findings

Minimal ribbonlengths determined for Salomon and Turk's head knots

Minimal ribbonlengths for granny and square knots calculated

Conjectures proposed relating ribbonlength to knot invariants

Abstract

The main open problem in geometric knot theory is to provide a tabulation of knots based on an energy criterion, with the goal of presenting this tabulation in terms of global energy minimisers within isotopy classes, often referred to as ideal knots. Recently, the first examples of minimal length diagrams and their corresponding length values have been determined by Ayala, Kirszenblat, and Rubinstein. This article is motivated by the scarcity of examples despite several decades of intense research. Here, we compute the minimal ribbonlength for some well-known knot diagrams, including the Salomon knot and the Turk's head knot. We also determine the minimal ribbonlength for the granny knot and square knot using a direct method. We conclude by providing the ribbonlength for infinite classes of critical ribbon knots, along with conjectures aimed at relating ribbonlength to knot invariants in pursuit of a metric classification of knots.