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

Abstract

Given a 2-crossing minimal chart $\Gamma$, a minimal chart with two crossings, set $\alpha=\min\{~i~|~$there exists an edge of label $i$ containing a white vertex$\}$, and $\beta=\max\{~i~|~$there exists an edge of label $i$ containing a white vertex$\}$. In this paper we study the structure of a neighbourhood of $\Gamma_\alpha\cup\Gamma_\beta$, and propose a normal form for 2-crossing minimal $n$-charts, here $\Gamma_\alpha$ and $\Gamma_\beta$ mean the union of all the edges of label $\alpha$ and $\beta$ respectively.