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 $\Gamma \cap \partial D\subset {\Gamma }_{\alpha +1}\cup {\Gamma }_{\beta -1}$.

39 pages, 19 figures

Cited by in corpus (1)

• Properties of minimal charts and their applications V: charts of type $(3,2,2)$