On the necessity of Reidemeister move 2 for simplifying immersed planar curves

TL;DR

This paper disproves Oestlund's 2001 conjecture by demonstrating that Reidemeister move 2 is necessary for simplifying immersed planar curves, showing that moves 1 and 3 alone are insufficient.

Contribution

The paper proves that Reidemeister move 2 cannot be omitted in the process of simplifying immersed planar curves, refuting a longstanding conjecture.

Findings

Reidemeister move 2 is essential for curve simplification

Moves 1 and 3 alone do not suffice for homotopy to the standard embedding

Disproof of Oestlund's conjecture from 2001

Abstract

In 2001, Oestlund conjectured that Reidemeister moves 1 and 3 are sufficient to describe a homotopy from any generic immersion from the circle into the plane to the standard embedding of the circle. We show that this conjecture is false.