  HD 3167 is a bright (V = 8.9), nearby K0 star observed by the NASA K2 mission
(EPIC 220383386), hosting two small, short-period transiting planets. Here we
present the results of a multi-site, multi-instrument radial velocity campaign
to characterize the HD 3167 system. The masses of the transiting planets are
5.02+/-0.38 MEarth for HD 3167 b, a hot super-Earth with a likely rocky
composition (rho_b = 5.60+2.15-1.43 g/cm^3), and 9.80+1.30-1.24 MEarth for HD
3167 c, a warm sub-Neptune with a likely substantial volatile complement (rho_c
= 1.97+0.94-0.59 g/cm^3). We explore the possibility of atmospheric composition
analysis and determine that planet c is amenable to transmission spectroscopy
measurements, and planet b is a potential thermal emission target. We detect a
third, non-transiting planet, HD 3167 d, with a period of 8.509+/-0.045 d
(between planets b and c) and a minimum mass of 6.90+/-0.71 MEarth. We are able
to constrain the mutual inclination of planet d with planets b and c: we rule
out mutual inclinations below 1.3 degrees as we do not observe transits of
planet d. From 1.3-40 degrees, there are viewing geometries invoking special
nodal configurations which result in planet d not transiting some fraction of
the time. From 40-60 degrees, Kozai-Lidov oscillations increase the system's
instability, but it can remain stable for up to 100Myr. Above 60 degrees, the
system is unstable. HD 3167 promises to be a fruitful system for further study
and a preview of the many exciting systems expected from the upcoming NASA TESS
mission.
