Hard Diagrams of the Unknot

TL;DR

This paper introduces three complex diagrams of the unknot that are difficult to simplify, requiring multiple crossings and computational effort, highlighting challenges in unknot diagram simplification.

Contribution

The paper constructs and analyzes three new hard unknot diagrams requiring at least three extra crossings to simplify, using computational methods to demonstrate their complexity.

Findings

Three hard unknot diagrams requiring ≥3 extra crossings

No small standard hard unknot diagram needs more than one crossing

Computational methods are essential for proving diagram complexity

Abstract

We present three "hard" diagrams of the unknot. They require (at least) three extra crossings before they can be simplified to the trivial unknot diagram via Reidemeister moves in $\mathbb{S}^2$. Both examples are constructed by applying previously proposed methods. The proof of their hardness uses significant computational resources. We also determine that no small "standard" example of a hard unknot diagram requires more than one extra crossing for Reidemeister moves in $\mathbb{S}^2$.