  We investigate the resolved kinematics of the molecular gas, as traced by
ALMA in CO (2-1), of 25 cluster member galaxies across three different clusters
at a redshift of $z\sim1.6$. This is the first large-scale analysis of the
molecular gas kinematics of cluster galaxies at this redshift. By separately
estimating the rotation curve of the approaching and receding side of each
galaxy via kinematic modeling, we quantify the difference in total circular
velocity to characterize the overall kinematic asymmetry of each galaxy. 3/14
of the galaxies in our sample that we are able to model have similar degrees of
asymmetry as that observed in galaxies in the field at similar redshift.
However, this leaved 11/14 galaxies in our sample with significantly higher
asymmetry, and some of these galaxies have degrees of asymmetry of up to
$\sim$50 times higher than field galaxies observed at similar redshift. Some of
these extreme cases also have one-sided tail-like morphology seen in the
molecular gas, supporting a scenario of tidal and/or ram pressure interaction.
Such stark differences in the kinematic asymmetry in clusters versus the field
suggest the evolutionary influence of dense environments, established as being
a major driver of galaxy evolution at low-redshift, is also active in the
high-redshift universe.
