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

Noboru Ito; Yusuke Takimura

doi:10.1016/j.topol.2016.07.008

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arXiv · 2011.14322

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

Noboru Ito·Yusuke Takimura

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Abstract

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.

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Noboru Ito, Yusuke Takimura. 2020-11-29. On a nontrivial knot projection under (1, 3) homotopy. https://doi.org/10.1016/j.topol.2016.07.008

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