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Crystallographic symmetries enforcing band touchings (BTs) in the Brillouinzone (BZ) have been utilized to classify and predict the topologicalsemimetals. Though the early proposed topological semimetals contain isolatednodal points in the BZ, the proposed nodal line semimetals later could hostvarious structures of several nodal lines/loops: nodal chains, nodal nets orHopf-links, etc. In this work, using compatibility relations, we first list allpossible high symmetry lines (HSLs) that can be nodal lines itself, highsymmetry planes (HSPLs) that can host nodal loops, high symmetry planes (HSPLs)that are nodal surfaces for all 230 SGs, with spin-orbit coupling andtime-reversal symmetry considered or not. We then show how to diagnose a nodalloop from the band crossing in an HSL, or nodal line/surface from irreduciblerepresentation (irrep) of an high-symmetry point (HSP), while the rest casescorrespond to nodal points. Among our results, those essential cases, for whichthe nodal points/lines/loops/surfaces must exist, are highlighted since theyare promising for the realizations of (nearly) ideal nodalpoint/line/loop/surface semimetals, as well as systems with flexible tunabilityowning fixed structure of topological nodal points/lines/loops/surfaces. Basedon our results, SGs allowing Hopf-link structure with one straight nodal linethreading a nodal loop, or two nesting nodal loops lying in two respective highsymmetry planes, are highlighted, with the predicted materials beingB$_5$Pb$_2$IO$_9$ in SG 34 and SrAl$_2$Au$_3$ in SG 62, respectively. Ourexhaustive results could serve as a useful guide for efficiently predicting anddesigning materials or artificial systems owning exotic geometric nodalstructures of energy bands simply based on structure symmetries.

Symmetry-enforced Band Nodes in 230 Space Groups