We predict linear polarization for a radioactively-powered kilonova following the merger of a black hole and a neutron star. Specifically, we perform 3-D Monte Carlo radiative transfer simulations for two different models, both featuring a lanthanide-rich dynamical ejecta component from numerical-relativity simulations while only one including an additional lanthanide-free disk wind component. We calculate polarization spectra for nine different orientations at 1.5, 2.5 and 3.5 d after the merger and in the $0.1-2,mu$m wavelength range. We find that both models are polarized at a detectable level 1.5 d after the merger while show negligible levels thereafter. The polarization spectra of the two models are significantly different. The model lacking a disk wind shows no polarization in the optical, while a signal increasing at longer wavelengths and reaching $sim1%-6%$ at $2,mu$m depending on the orientation. The model with a disk-wind component, instead, features a characteristic double-peak polarization spectrum with one peak in the optical and the other in the infrared. Polarimetric observations of future events will shed light on the debated neutron richness of the disk-wind component. The detection of optical polarization would unambiguously reveal the presence of a lanthanide-free disk-wind component, while polarization increasing from zero in the optical to a peak in the infrared would suggest a lanthanide-rich composition for the whole ejecta. Future polarimetric campaigns should prioritize observations in the first $sim48$ hours and in the $0.5-2,mu$m range, where polarization is strongest, but also explore shorter wavelengths/later times where no signal is expected from the kilonova and the interstellar polarization can be safely estimated.
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