  The nuclear symmetry energy plays a key role in determining the equation of
state of dense, neutron-rich matter, which governs the properties of both
terrestrial nuclear matter as well as astrophysical neutron stars. A recent
measurement of the neutron skin thickness from the PREX collaboration has lead
to new constraints on the slope of the nuclear symmetry energy, L, which can be
directly compared to inferences from gravitational-wave observations of the
first binary neutron star merger inspiral, GW170817 In this paper, we explore a
new regime for potentially constraining the slope, L, of the nuclear symmetry
energy with future gravitational wave events: the post-merger phase a binary
neutron star coalescence. In particular, we go beyond the inspiral phase, where
imprints of the slope parameter L may be inferred from measurements of the
tidal deformability, to consider imprints on the post-merger dynamics,
gravitational wave emission, and dynamical mass ejection. To this end, we
perform a set of targeted neutron star merger simulations in full general
relativity using new finite-temperature equations of state, which
systematically vary L. We find that the post-merger dynamics and gravitational
wave emission are mostly insensitive to the slope of the nuclear symmetry
energy. In contrast, we find that dynamical mass ejection contains a weak
imprint of L, with large values of L leading to systematically enhanced ejecta.
