Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle

TL;DR

This paper constructs unknot diagrams that require a quadratic number of Reidemeister moves to untangle, demonstrating that the complexity of untangling can grow quadratically with the number of crossings.

Contribution

It provides explicit examples of unknot diagrams with quadratic lower bounds on Reidemeister moves needed for untangling, advancing understanding of knot complexity.

Findings

Unknot diagrams requiring quadratic Reidemeister moves to untangle

Bounds apply in both $S^2$ and $ ^2$

Demonstrates quadratic complexity in unknot untangling

Abstract

We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in $S^2$ and in $\R^2$.