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

In 2001, Östlund 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 (Östlund conjecture)....

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Bibliographic Details
Published in:Topology and its applications 2016-09, Vol.210, p.22-28
Main Authors: Ito, Noboru, Takimura, Yusuke
Format: Article
Language:English
Subjects:
Knot projections
Reidemeister moves
Spherical curves
Östlund conjecture
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Summary:In 2001, Östlund 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 (Östlund 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.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2016.07.008