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

TL;DR

This paper investigates the structural properties of minimal n-charts with two crossings, focusing on the configuration of complementary domains and their relation to specific label unions, as a foundational step for enumeration.

Contribution

It introduces a detailed analysis of the structure of minimal charts with two crossings, particularly the behavior of certain disks and label unions, advancing the classification effort.

Findings

Crossings are contained in the intersection of specific label unions.

Disks not containing crossings have boundary intersections limited to certain label sets.

Structural constraints help in enumerating minimal charts with two crossings.

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