Complexity of plane and spherical curves

Abstract

We show that the maximal number of singular moves required to pass between any two regularly homotopic plane or spherical curves with at most $n$ crossings grows quadratically with respect to $n$. Furthermore, for any two regularly homotopic curves with at most $n$ crossings, there exists such a sequence of singular moves, satisfying the quadratic bound, for which all curves along the way have at most $n+2$ crossings