Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Classification of doubly periodic untwisted (p,q)-weaves by their
  crossing number   A weave is the lift to the Euclidean thickened plane of a set of infinitely
many planar crossed geodesics, that can be characterized by a number of sets of
threads describing the organization of the non-intersecting curves, together
with a set of crossing sequences representing the entanglements. In this paper,
the classification of a specific class of doubly periodic weaves, called
untwisted (p,q)-weaves, is done by their crossing number, which is the minimum
number of crossings that can possibly be found in a unit cell of its infinite
weaving diagrams. Such a diagram can be considered as a particular type of
quadrivalent periodic planar graph with an over or under information at each
vertex, whose unit cell corresponds to a link diagram in a thickened torus.
Moreover, considering that a weave is not uniquely defined by its sets of
threads and its crossing sequences, we also specify the notion of equivalence
classes by introducing a new parameter, called crossing matrix.

35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
16
The categories of this paper are: stat
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg