Classification of Exceptional Nodal Topologies Protected by $\mathcal{PT}$ Symmetry

Exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, and parity-time ($\mathcal{PT}$) symmetry, reflecting balanced gain and loss in photonic systems, are paramount concepts in non-Hermitian systems. We here complete the topological classification of exceptional nodal degeneracies protected by $\mathcal{PT}$ symmetry in up to three dimensions and provide simple example models whose exceptional nodal topologies include previously overlooked possibilities such as second-order knotted surfaces of arbitrary genus, third-order knots and fourth-order points.