Eigenvalue knots and their isotopic equivalence in three-state non-Hermitian systems

The spectrum of a non-Hermitian system generically forms a two-dimensional complex Riemannian manifold with a distinct topology from the underlying parameter space. This spectral topology permits eigenvalue trajectories to braid into knots. In this paper, through the analyses of encircling loops, exceptional points, and their topology, we uncover the necessary considerations for constructing eigenvalue knots and establish their relation to the spectral topology. Using an acoustic system with two periodic synthetic dimensions, we experimentally realize several knots with braid index 3. In addition, by highlighting the role of branch cuts on the eigenvalue manifolds, we show that eigenvalue knots produced by homotopic parametric loops are isotopic such that they can deform into one another by type-II or III Reidemeister moves. Our results not only provide a general recipe for constructing eigenvalue knots but also expand the current understanding of eigenvalue knots by showing that they contain information beyond that of the spectral topology.