Minimal charts of type (3,3)

Abstract

Let $\Gamma$ be a chart. For each label $m$, we denote by $\Gamma_m$ the "subgraph" of $\Gamma$ consisting of all the edges of label $m$ and their vertices. Let $\Gamma$ be a minimal chart of type $(m;3,3)$. That is, a minimal chart $\Gamma$ has six white vertices, and both of $\Gamma_m\cap\Gamma_{m+1}$ and $\Gamma_{m+1}\cap\Gamma_{m+2}$ consist of three white vertices. Then $\Gamma$ is C-move equivalent to a minimal chart containing a "subchart" representing a 2-twist spun trefoil or its "reflection".