  The flat-spectrum radio quasar 3C 345 has been showing gamma-ray activity
since the mid-2000s along with activity across the electromagnetic spectrum. A
gamma-ray burst in 2009 was successfully linked to relativistic outflow in 43
GHz VLBI observations and has since been analyzed also using single dish
measurements. A multi-wavelength follow-up VLBI observation to the 2009 flare
in conjunction with 43 GHz catalogue data from the VLBA-BU-BLAZAR and BEAM-ME
programs are analyzed in this study in the context of the long-term evolution
of the source. We aim to probe the innermost few milli-arcseconds of the
ultracompact 3C 345 jet. In the process, we analyze the long-term kinematics of
the inner jet and discuss the magnetic field morphology at different scales, as
well as the origin of the gamma-ray emission. New observations at 23, 43, and
86 GHz took place on ten epochs between 2017 and 2019. We calibrate the 30 data
sets using the rPicard pipeline, image them in Difmap and carry out
polarization calibration using the GPCAL pipeline. We complement our VLBI data
with ancillay VLBI maps at multiple frequencies, as well as 43 GHz observations
carried out in the framework of the BEAM-ME and VLBA-BU-BLAZAR monitoring
programs. We find multiple distinct component paths in the inner jet, forming a
helical geometry. The helix is anchored at a stationary feature some 0.16 mas
from the 43 GHz VLBI core and has a timescale of about 8 years. The
characteristic bends in the jet morphology are caused by variations in the
component ejection angle. We confirm the result of previous studies that the
gamma-ray emission is produced (or caused) by relativistic outflow and violent
interactions within the jet.
