Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The structure of a minimal $n$-chart with two crossings II:
  Neighbourhoods of $\Gamma_1\cup\Gamma_{n-1}$   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.

28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics