I.B. thanks N. Lvov and Z. Szabó for discussions on linking numbers. We thank D. Lu and M. Hashimoto at Beamline 5-2 of the Stanford Synchrotron Radiation Lightsource at the SLAC National Accelerator Laboratory, CA, USA for support. I.B. and D.M. thank T. Muro for experimental support during preliminary ARPES measurements carried out at BL25SU of SPring-8 in Hyogo, Japan. I.B. thanks B. Lian for discussions on the topological magneto-electric effect. I.B., T.A.C., X.P.Y. and D.M. thank J. McChesney and F. Rodolakis for experimental support during preliminary ARPES measurements carried out at BL29 of the Advanced Photon Source in Illinois, USA. I.B. acknowledges discussions with B. Belopolski on Savitzky–Golay analysis. G. Chang acknowledges the support of the National Research Foundation, Singapore under its NRF Fellowship Award (NRF-NRFF13-2021-0010) and the Nanyang Assistant Professorship grant from Nanyang Technological University. T.A.C. acknowledges support by the National Science Foundation Graduate Research Fellowship Program under grant number DGE-1656466. A.C. acknowledges funding from the Swiss National Science Foundation under grant number 200021-165529. We acknowledge synchrotron radiation beamtime at the ADRESS beamline of the Swiss Light Source of the Paul Scherrer Institut in Villigen, Switzerland under proposals 20170898, 20190740 and 20191674. S.-M.H. acknowledges funding by the MOST-AFOSR Taiwan program on Topological and Nanostructured Materials under grant no. 110-2124-M-110-002-MY3. We further acknowledge use of Princeton’s Imaging and Analysis Center, which is partially supported by the Princeton Center for Complex Materials, a National Science Foundation Materials Research Science and Engineering Center (DMR-2011750). This research used resources of the Advanced Photon Source, a US Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under contract number DE-AC02-06CH11357. We acknowledge beamtime at BL25SU of SPring-8 under proposal 2017A1669 and at BL29 of the Advanced Photon Source under proposals 54992 and 60811. K.M. and C.F. acknowledge financial support from the European Research Council Advanced Grant no. 742068 “TOP-MAT”. C.F. acknowledges the DFG through SFB 1143 (project ID. 247310070) and the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC2147, project ID. 39085490). M.Z.H. acknowledges support from the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center and Princeton University. M.Z.H. acknowledges visiting scientist support at Berkeley Lab (Lawrence Berkeley National Laboratory) during the early phases of this work. Work at Princeton University was supported by the Gordon and Betty Moore Foundation (grant numbers GBMF4547 and GBMF9461; M.Z.H.). The ARPES and theoretical work were supported by the US DOE under the Basic Energy Sciences programme (grant number DOE/BES DE-FG-02-05ER46200; M.Z.H.). Use of the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, is supported by the US DOE, Office of Science, Office of Basic Energy Sciences, under contract number DE-AC02-76SF00515. We acknowledge MAX IV Laboratory for time on the BLOCH Beamline under proposal 20210268. Research conducted at MAX IV, a Swedish national user facility, is supported by the Swedish Research council under contract 2018-07152, the Swedish Governmental Agency for Innovation Systems under contract 2018-04969, and Formas under contract 2019-02496. Materials characterization and the study of topological quantum properties were supported by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center and Princeton University.