  Topological semimetals with band crossings (BCs) near the Fermi level have
attracted intense research activities in the past several years. Among various
BCs, those enforced by an hourglass-like connectivity pattern, which are just
located at the vertex in the neck of an hourglass and thus called hourglass BCs
(HBCs), show interesting topological properties and are intimately related with
the space group symmetry. Through checking compatibility relations in the
Brillouin zone (BZ), we list all possible HBCs for all 230 space groups by
identifying positions of HBCs as well as the compatibility relations related
with the HBCs.The HBCs can be coexisting with conventional topological BCs such
as Dirac andWeyl fermions and based on our exhaustive list, the dimensionality
and degeneracy of the HBCs can be quickly identified. It is also found that the
HBCs can be classified into two categories: one contains essential HBCs which
are guaranteed to exist, while the HBCs in the other category may be tuned to
disappear. Our results can help in efficiently predicting hourglass semimetals
combined with first-principles calculations as well as studying transitions
among various topological crystalline phases.
