The structure of a minimal $n$-chart with two crossings II: Neighbourhoods of $Γ_1\cupΓ_{n-1}$

Abstract

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