Exploring four-dimensional topological manifolds of non-Hermitian bands from Klein bottles to spun trefoil knots

Non-Hermiticity is prevalent across various branches of physics, giving rise to a range of unique phenomena with no Hermitian counterparts. Recently, the knot topology and braiding properties of non-Hermitian bands have been experimentally observed in systems such as coupled ring resonators with synthetic dimensions, cavity optomechanical systems, and more. However, these observations have been limited to one-dimensional topological manifolds (string knots) in three-dimensional (3D) space, leaving unanswered the question of whether higher-dimensional topological manifolds in non-Hermitian bands can be realized. In this work, we present the construction of two distinct four-dimensional (4D) topological manifolds - the Klein bottle and spun trefoil knot - formed by two-dimensional (2D) complex non-Hermitian bands in synthetic energy space. Moreover, the sinusoidal torus and self-intersected Roman surface can appear in the 3D projection of these 4D non-Hermitian topological manifolds. Experimentally, we design and fabricate reconfigurable electric circuits to reconstruct 2D non-Hermitian bands and observe these non-Hermitian 4D topological manifolds. The strong alignment between our theoretical predictions and experimental results paves the way for generating surface topologies in non-Hermitian bands and opens avenues for exploring high-dimensional topological manifolds in artificial systems.