On the necessity of Reidemeister move 2 for simplifying immersed planar   curves
  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. 
Tobias J. Hagge, Jonathan T. Yazinski