On a nontrivial knot projection under (1,3) homotopy

In 2001, Ostlund formulated the question: are Reidemeister moves of types 1 and 3 sufficient to describe a homotopy from any generic immersion of a circle in a two-dimensional plane to an embedding of the circle? The positive answer to this question was treated as a conjecture (Ostlund conjecture). In 2014, Hagge and Yazinski disproved the conjecture by showing the first counterexample with a minimal crossing number of 16. This example is naturally extended to counterexamples with given even minimal crossing numbers more than 14. This paper obtains the first counterexample with a minimal crossing number of 15. This example is naturally extended to counterexamples with given odd minimal crossing numbers more than 13. (C) 2016 Elsevier B.V. All rights reserved.