Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Geometrically incompressible non-orientable closed surfaces in lens
  spaces   We consider non-orientable closed surfaces of minimum crosscap number in the
$(p,q)$-lens space $L(p,q) \cong V_1 \cup_{\partial} V_2$, where $V_1$ and
$V_2$ are solid tori.
  Bredon and Wood gave a formula for calculating the minimum crosscap number.
  Rubinstein showed that $L(p,q)$ with $p$ even has only one isotopy class of
such surfaces, and it is represented by a surface in a standard form, which is
constructed from a meridian disk in $V_1$ by performing a finite number of band
sum operations in $V_1$ and capping off the resulting boundary circle by a
meridian disk of $V_2$.
  We show that the standard form corresponds to an edge-path $\lambda$ in a
certain tree graph in the closure of the hyperbolic upper half plane.
  Let $0=p_0/q_0, p_1/q_1, ..., p_k/q_k = p/q$ be the labels of vertices which
$\lambda$ passes.
  Then the slope of the boundary circle of the surface right after the $i$-th
band sum is $(p_i, q_i)$.
  The number of edges of $\lambda$ is equal to the minimum crosscap number.
  We give an easy way of calculating $p_i / q_i$ using a certain continued
fraction expansion of $p/q$.

5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
7
The categories of this paper are: hep-th
22
The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph