Measuring the Trefoil Knot of Degneracies Around a Triple Exceptional Point

When a non-Hermitian system's Ν×Ν dynamical matrix (or "Hamiltonian") is tuned around a loop that does not intersect any degeneracies, the system's Ν complex eigenvalues trace out a braid. We show that the specific braid produced by a given loop is determined by how that loop encloses the system's double degeneracies (ΕΡ₂'s). We have measured the eigenvalue spectra of a three-mode mechanical system (which is tuned via standard optomechanical techniques), and show that the system's triple degeneracy point (ΕΡ₃) is surrounded by a knotted structure of ΕΡ₂'s. This structure is closely related to the well-known trefoil knot. These measurements (and the measurements of eigenvalue braids presented in the preceding talk) agree well with calculations based on this specific device's optomechanical properties. More importantly, this work illustrates how concepts from algebraic geometry (e.g., knots and braids) can play a useful role in understanding the eigenvalue topology of non-Hermitian systems.