  We investigate an antiferromagnetic spin-$1$ Heisenberg chain in the presence
of Dyzaloshinskii-Moriya interactions (DMI) and an external magnetic field. We
study the resulting spin chain using a combination of numerical and analytical
techniques. Using DMRG simulations to determine the spectral gap and the
entanglement spectrum, we map out the phase diagram as a function of magnetic
field strength and DMI strength. We provide a qualitative interpretation for
these numerical findings by mapping the spin-$1$ chain on a spin-$1/2$ ladder
and using a bosonization approach.
