  As the counterpart of Hermitian nodal structures, the geometry formed by
exceptional points (EPs), such as exceptional lines (ELs), entails intriguing
spectral topology. We report the experimental realization of order-3
exceptional lines (EL3) that are entirely embedded in order-2 exceptional
surfaces (ES2) in a three-dimensional periodic synthetic momentum space. The
EL3 and the concomitant ES2, together with the topology of the underlying
space, prohibit the evaluation of their topology in the eigenvalue manifold by
prevailing topological characterization methods. We resolve this issue by
defining a winding number that associates with the resultants of the
Hamiltonian. This resultant winding number detects EL3 but ignores the ES2,
allowing the diagnosis of the topological currents carried by the EL3, which
enables the prediction of their evolution under perturbations. Our results
exemplify unprecedented topology of higher-order exceptional geometries and may
inspire new non-Hermitian topological applications.
