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

Volume 103 / 2014

Tobias Hagge, Jonathan Yazinski Banach Center Publications 103 (2014), 101-110 MSC: Primary 14H50. DOI: 10.4064/bc103-0-4


In 2001, motivated by his results on finite-type knot diagram invariants, Östlund conjectured that Reidemeister moves 1 and 3 are sufficient to describe a homotopy from any generic immersion $S^{1} \rightarrow\mathbb{R}^{2}$ to the standard embedding of the circle. We show that this conjecture is false.


  • Tobias HaggeThe University of Texas at Dallas
    FO 35, 800 West Campbell Rd
    Richardson, TX 75080, USA
  • Jonathan YazinskiDepartment of Mathematics
    College of William & Mary
    P.O. Box 8795
    Williamsburg, VA 23187-8795, USA

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