On charts with two crossings II

Abstract

Let $\Gamma$ be a chart with at most two crossings. In this paper, we show that if $\Gamma$ is a 2-minimal generalized $n$-chart with $n \ge 5$, then $\Gamma$ contains at least $4n-10$ black vertices. And we show that if the closure of the surface braid represented by $\Gamma$ is a disjoint union of spheres, then $\Gamma$ is a ribbon chart. Hence the closure is a ribbon surface.