  Real world systems comprised of coupled oscillators have the ability to
exhibit spontaneous synchronization and other complex behaviors. The interplay
between the underlying network topology and the emergent dynamics remains a
rich area of investigation for both theory and experiment. In this work we
study lattices of coupled Kuramoto oscillators with non-local interactions. Our
focus is on the stability of twisted states. These are equilibrium solutions
with constant phase shifts between oscillators resulting in spatially linear
profiles. Linear stability analysis follows from studying the quadratic form
associated with the Jacobian matrix. Novel estimates on both stable and
unstable regimes of twisted states are obtained in several cases. Moreover,
exploiting the "almost circulant" nature of the Jacobian obtains a surprisingly
accurate numerical test for stability. While our focus is on 2D square
lattices, we show how our results can be extended to higher dimensions.
