  Crystallographic symmetries enforcing band touchings (BTs) in the Brillouin
zone (BZ) have been utilized to classify and predict the topological
semimetals. Though the early proposed topological semimetals contain isolated
nodal points in the BZ, the proposed nodal line semimetals later could host
various structures of several nodal lines/loops: nodal chains, nodal nets or
Hopf-links, etc. In this work, using compatibility relations, we first list all
possible high symmetry lines (HSLs) that can be nodal lines itself, high
symmetry planes (HSPLs) that can host nodal loops, high symmetry planes (HSPLs)
that are nodal surfaces for all 230 SGs, with spin-orbit coupling and
time-reversal symmetry considered or not. We then show how to diagnose a nodal
loop from the band crossing in an HSL, or nodal line/surface from irreducible
representation (irrep) of an high-symmetry point (HSP), while the rest cases
correspond to nodal points. Among our results, those essential cases, for which
the nodal points/lines/loops/surfaces must exist, are highlighted since they
are promising for the realizations of (nearly) ideal nodal
point/line/loop/surface semimetals, as well as systems with flexible tunability
owning fixed structure of topological nodal points/lines/loops/surfaces. Based
on our results, SGs allowing Hopf-link structure with one straight nodal line
threading a nodal loop, or two nesting nodal loops lying in two respective high
symmetry planes, are highlighted, with the predicted materials being
B$_5$Pb$_2$IO$_9$ in SG 34 and SrAl$_2$Au$_3$ in SG 62, respectively. Our
exhaustive results could serve as a useful guide for efficiently predicting and
designing materials or artificial systems owning exotic geometric nodal
structures of energy bands simply based on structure symmetries.

Materials Science