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

TL;DR

This paper constructs the first minimal crossing number 15 counterexample to Ostlund's conjecture, showing Reidemeister moves of types 1 and 3 are insufficient for certain homotopies of circle immersions.

Contribution

It provides the first minimal crossing number 15 counterexample to the Ostlund conjecture, extending previous counterexamples to odd crossing numbers.

Findings

Counterexample with minimal crossing number 15 constructed.

Counterexamples extended to odd minimal crossing numbers >13.

Reidemeister moves of types 1 and 3 are not sufficient for all homotopies.

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.