RII number of knot projections

Abstract

Every knot projection is simplified to the trivial spherical curve not increasing double points by using deformations of types 1, 2, and 3 which are analogies of Reidemeister moves of types 1, 2, and 3 on knot diagrams. We introduce RII number of a knot projection that is the minimum number of deformations of negative type 2 among such sequences. By definition, it is invariant under deformations of types 1 and 3. This is motivated by \"{O}stlund conjecture: Deformations of type 1 and 3 are sufficient to describe a homotopy from any generic immersion of a circle in a two dimensional plane to an embedding of the circle (2001), which implies RII number always would be zero. However, Hagge and Yazinski disproved the conjecture by showing the first counterexample with 16 double points, which implies that RII number is nontrivial. This paper shows that RII number can be any nonnegative number.