Higher order topological knot and complex Chern insulator

We present "higher-order topological knot" (HOTK) phases, new non-Hermitian higher-order topological phases in class AI of the 38-fold classification. These phases are protected by crystalline symmetries and bulk line gaps, yet under open boundary conditions, they possess topological edge states with unidirectional propagation protected by point gaps. Upon breaking the protecting crystalline symmetries, the edge states of HOTK phases can exhibit a higher-order skin effect. Notably, HOTK phases can transition into "Complex Chern insulator" phases, which posses bulk bands with nonzero Chern numbers protected by imaginary lines gaps in the presence of time-reversal symmetry.