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 very long baseline interferometry (VLBI) observations and has since also been analyzed using single-dish measurements. A multiwavelength follow-up VLBI observation of the 2009 flare in conjunction with 43\,GHz catalog 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 milliarcseconds of the ultracompact 3C\,345 jet. To do so, we analyzed 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 were carried out at ten epochs between 2017 and 2019. We calibrated the 30 datasets using the rPicard pipeline, imaged them in Difmap and carried out polarization calibration using the GPCAL pipeline.
We complemented our VLBI data by 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, which together have a helical geometry. The helix appears to be anchored at a stationary feature some 0.16\,mas from the 43\,GHz VLBI core and has an evolution 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: the gamma-ray emission is caused by relativistic outflow and violent interactions within the jet.