  The intrinsic alignments of galaxies, i.e., the correlation between galaxy
shapes and their environment, are a major source of contamination for weak
gravitational lensing surveys. Most studies of intrinsic alignments have so far
focused on measuring and modelling the correlations of luminous red galaxies
with galaxy positions or the filaments of the cosmic web. In this work, we
investigate alignments around cosmic voids. We measure the intrinsic alignments
of luminous red galaxies detected by the Sloan Digital Sky Survey around a
sample of voids constructed from those same tracers and with radii in the
ranges: $[20-30; 30-40; 40-50]$ $h^{-1}$ Mpc and in the redshift range
$z=0.4-0.8$. We present fits to the measurements based on a linear model at
large scales, and on a new model based on the void density profile inside the
void and in its neighbourhood. We constrain the free scaling amplitude of our
model at small scales, finding no significant alignment at $1\sigma$ for either
sample. We observe a deviation from the null hypothesis, at large scales, of
2$\sigma$ for voids with radii between 20 and 30 $h^{-1}$ Mpc, and 1.5 $\sigma$
for voids with radii between 30 and 40 $h^{-1}$ Mpc and constrain the amplitude
of the model on these scales. We find no significant deviation at 1$\sigma$ for
larger voids. Our work is a first attempt at detecting intrinsic alignments of
galaxy shapes around voids and provides a useful framework for their mitigation
in future void lensing studies.
