Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Topological carbon allotropes: paradigm shift for materials innovation   Topology is a central concept of mathematics, which allows us to distinguish
two isolated rings with linked ones. In material science, researchers
discovered topologically different carbon allotropes in a form of a cage, a
tube, and a sheet, which have unique translational and rotational symmetries,
described by a crystallographic group theory, and the atoms are arranged at
specific rigid positions in 3-dimensional ($D$) space. However, topological
orders must be robust against deformations, so that we can make completely
different families of topological materials. Here we propose various
topological structures such as knots and links using covalent $\sigma$ bonds of
carbon atoms, while allowing various topologically equivalent arrangements
using weak $\pi$ bonds. By extending this idea, we invented a new 3D carbon
allotrope, Hopfene, which has periodic arrays of Hopf-links to knit horizontal
Graphene sheets into vertical ones without connecting by $\sigma$ bonds.

18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga