  Among the most intriguing features of non-Hermitian (NH) systems is the
capability of their complex energies to form braids under parametric variation.
Several braiding behaviors, including link and knot formation, have been
observed in experiments on synthetic NH systems such as looped optical fibers.
Though the exact conditions for these phenomena remain unsettled, existing
demonstrations have involved long-range nonreciprocal hoppings, which are hard
to implement on many experimental platforms. Here, we introduce a route to
realize complex energy braids using 1D NH Aubry-Andre-Harper lattices. Under
purely local gain and loss modulation, the eigenstates exhibit a variety of
different braiding behaviors, including unknots, Hopf links, trefoil knots,
Solomon links and catenanes. We show how these are created by the interplay
between non-Hermiticity and the lattice bulk states and topological edge
states. The transitions between different braiding configurations are marked by
changes in the global Berry phase of the NH lattice. These findings reveal
interesting connections between the braiding of complex energies and NH band
topology.
