The structure of a minimal $n$-chart with two crossings I: Complementary domains of $\Gamma_1\cup\Gamma_{n-1}$

Abstract

This is the first step of the two steps to enumerate the minimal charts with two crossings. For a label $m$ of a chart $\Gamma$ we denote by $\Gamma_m$ the union of all the edges of label $m$ and their vertices. For a minimal chart $\Gamma$ with exactly two crossings, we can show that the two crossings are contained in $\Gamma_\alpha\cap\Gamma_\beta$ for some labels $\alpha<\beta$. In this paper, we study the structure of a disk $D$ not containing any crossing but satisfying $\Gamma\cap \partial D\subset\Gamma_{\alpha+1}\cup \Gamma_{\beta-1}$.