Fig. 2

Zero-energy states under open boundary conditions along the \(\hat{w}\) direction for various 4D nodal loop systems. a–c Boundary states of a single 4D nodal loop of (Eq. (2)) with (a) and without (b, c) the perturbation term \({h}_{0}({\bf{k}}){\mathbb{I}}\), with \({\mathbb{I}}\) the identity matrix acting on a pesudospin-1/2 space. The form of the perturbation is chosen to be \({h}_{0}({\bf{k}})=0.4\cos {k}_{y}\) in (b), and \({h}_{0}({\bf{k}})=0.4\sin {k}_{y}\) in (c), demonstrating the robustness of the Seifert surface against such perturbations. Red regions represent zero-energy bulk states (nodal solutions) whereas dark and light blue regions depict boundary states as Seifert surfaces of the nodal loops. Note that in (c), the light blue and dark blue regions are partially covered up by each other. d Boundary Trefoil knot and its Seifert surface states from Eq. (4). e, f Boundary Hopf-link and Borromean rings given by Eq. (8) and their Serfiet surfaces, with \(N=2\) and \(N=3\) linked loops respectively. \({k}_{x,y,z}\) represent the first three quasi-momenta of the boundary Brillouin zone with open boundary conditions along the fourth \(\hat{w}\) direction