Fig. 1

3D and 4D nodal loops and their corresponding boundary states. a A Hopf-link in a 3D nodal loop (NL) system, and b the corresponding 2D boundary Brillouin zone (BZ) under open boundary conditions (OBCs) along \(z\) direction, with yellow and blue curves corresponding to projected NLs and brown shaded regions corresponding to topological boundary states. The Hopf-link in a 3D NL system gives rise to 2D topological drumhead states (dark and light brown) upon \(\hat{z}\)-boundary projection, retaining no information about the 3D over/under-crossings. c Two isolated NLs in a 4D system and d a Trefoil knot in a 4D system, with (e, f) their 3D boundary BZ under OBCs along \(\hat{w}\) direction. Full information on the knot/link topology is retained in the Seifert surfaces [brown in (e–f)] arising from topological states in the 3D boundaries of 4D NL systems (c–d). A two-component boundary Hopf-link (e) can arise from a 4D NL system with two unlinked single loops (c) indexed by different \({k}_{w}={k}_{1,2}\), while a one-component boundary Trefoil knot (f) and its Seifert surface also arises from a different 4D NL system (d). Here \({k}_{x,y,z,w}\) represent the quasi-momenta of 3D and 4D momentum spaces