  We analyzed data in polarized intensity obtained with the Very Long Baseline
Array (VLBA) at twelve epochs (one observation per month from January to
December 2011) at 15, 24, and 43 GHz. For the absolute orientation of the
electric vector position angles (EVPA) we used the D-terms method. We also used
gamma-ray data from the Fermi Large Area Telescope on weekly time bins
throughout 2011. The source shows polarized emission, and its properties vary
with time, frequency, and location along the jet. The core mean polarization
fraction is generally between 1% and 2%, with a 4% peak at 43 GHz in March; the
polarization angle is variable, mainly at 15 GHz, where it changes frequently,
and less so at 43 GHz, where it oscillates in the range 114 - 173 deg. The jet
polarization properties are more stable, with a fractional polarization of
around 16% and a polarization angle nearly perpendicular to the jet axis. The
average flux and photon index at gamma-ray energies are (17.7+-0.5)x10^8 ph
cm^-2 s^-1 and the photon index is 1.77+-0.02. The gamma-ray light curve shows
variability, with a main peak that appears to be associated with the peak in
the core polarized emission at 43 GHz, as well as with the total intensity
light curve. A discrete correlation function analysis yields a correlation
coefficient of 0.54 at zero delay, with a significance level above 99.7%. We
accurately determine the polarization properties of Mrk 421, both in the core
and in the jet region. The radio and gamma-ray light curves are correlated. The
observed EVPA variability at 15 GHz is partly due to opacity and partly to a
variable Faraday rotation effect. To explain the residual variability of the
intrinsic polarization angle and the low degree of polarization in the core
region, we invoke a blend of variable cross-polarized subcomponents with
different polarization properties within the beam.
