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.

arXiv comment57 pages, 36 figures